DETECTING PRO-p-GROUPS THAT ARE NOT ABSOLUTE GALOIS GROUPS, EXPANDED VERSION

DAVE BENSON, NICOLE LEMIRE, JÁN MINÁČ, AND JOHN SWALLOW

Let p be a prime. It is a fundamental problem to classify the absolute Galois groups G_{F} of fields F containing a primitive pth root of unity. In this paper we present several constraints on such G_{F}, using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these G_{F}. In the case p > 2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. In section 3 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of G_{F}. Finally, in sections 4–6 we construct new families of pro-p-groups which are not absolute Galois groups over any field F.

As a consequence of our results, we prove the following limitations on relator shapes of pro-p absolute Galois groups. For elements σ and τ of a group Γ, let ^{0}[σ, τ] = τ, ^{1}[σ, τ] = στσ^{−1}τ^{−1}, and ^{n}[σ, τ] = [σ,^{n−1}[σ, τ]] for n ≥ 2. Similarly, for subsets Γ_{1} and Γ_{2} of Γ, let ^{n}[Γ_{1}, Γ_{2}] denote the closed subgroup generated by all elements of the form ^{n}[γ_{1}, γ_{2}] for γ_{i} ∈ Γ_{i}.

Theorem A.1. Let p be an odd prime, Γ a pro-p-group with maximal closed subgroup Δ, and σ ∈ Γ \ Δ.

(1) Suppose that for some τ ∈ Δ and some 2 ≤ e ≤ p − 2

e[σ, τ] ∉ ^{p−1}[σ, Δ]Φ(Δ)
e+1[σ, τ] ∈ Φ(Δ).

Date: October 20, 2006.

Nicole Lemire was supported in part by Natural Sciences and Engineering Research Council of Canada grant R3276A01. Ján Mináč was supported in part by Natural Sciences and Engineering Research Council of Canada grant R0370A01.

John Swallow was supported in part by National Science Foundation grant DMS-0600122.

2 DAVE BENSON, NICOLE LEMIRE, JÁN MINÁČ, AND JOHN SWALLOW

Then Γ is not an absolute Galois group.

Moreover, if Γ contains a normal subgroup Λ ⊂ Δ such that Γ/Λ ≃ ℤ/p^{2}ℤ, we may take 1 ≤ e ≤ p − 2.

(2) Suppose that for some τ_{1}, τ_{2} ∈ Δ,

[σ, τ_{i}] ∉ ^{p−1}[σ, Δ]Φ(Δ),  i = 1, 2
2[σ, τ_{i}] ∈ Φ(Δ),  i = 1, 2
⟨[σ, τ_{1}]⟩Φ(Δ) ≠ ⟨[σ, τ_{2}]⟩Φ(Δ)

Then Γ is not an absolute Galois group.

(3) Suppose that

σ^{p} ∈ ^{2}[σ, Δ]Φ(Δ).

Then Γ is not an absolute Galois group.

Here Φ(Δ) = Δ^{p}[Δ, Δ] denotes the Frattini subgroup of Δ.

A striking corollary is that one-relator pro-p-groups with relations quite similar to the relations of Demuškin groups for odd primes [De] cannot be absolute Galois groups.

Corollary. Let p be an odd prime and Γ a pro-p-group minimally generated by {σ_{1}, σ_{2}} ∪ {σ_{i}}_{i∈I} subject to a single relation

σ_{1}^{q} ·^{f}[σ_{1}, σ_{2}] · ∏_{(i,j)∈J}[σ_{i}, σ_{j}] · ∏_{k∈K}[σ_{1}^{p}, σ_{k}]

for some 2 ≤ f ≤ p − 1, q ∈ ℕ ∪ {0} with q = 0 mod (p^{2}), a finite ordered set of pairs J ⊂ I × I, and a finite ordered subset K of I. Then Γ is not an absolute Galois group.

The results in [LMS] may be used to establish further new results on possible V -groups of fields and metabelian quotients of absolute Galois pro-p-groups. (For the definition of the V -group V_{F} of a field F, see section 3.) Moreover, some of the results here hold in a greater generality than their formulations here. For instance, the examples here of pro-p non-absolute Galois groups are also examples of groups which are not maximal pro-p-quotients of absolute Galois groups, by [LMS, §6]. Furthermore, pro-p-groups which are not absolute Galois groups are not p-Sylow subgroups of absolute Galois groups. We plan a systematic study of these concerns in [BMS].

DETECTING PRO-p-GROUPS 3

We observe that because this paper is concerned only with degree 1 and degree 2 cohomology, the results cited from [LMS] rely only on the Merkurjev-Suslin Theorem [MeSu, Theorem 11.5] and not the recent work on the full Bloch-Kato Conjecture.

1. T -groups

A T -group is a nontrivial pro-p-group T with a maximal closed subgroup N that is abelian of exponent dividing p. Then N is a normal subgroup, and the factor group T/N acts naturally on N: choose a lift t ∈ T and act via n ↦ tnt^{−1}. Given any profinite group Γ with a closed normal subgroup Δ of index p, the factor group Γ/Δ^{p}[Δ, Δ] is a T -group.

Now suppose that E/F is a cyclic field extension of degree p. We define the T -group of E/F to be T_{E/F} := G_{F} /G_{E}^{p}[G_{E}, G_{E}]. In this section we classify those T -groups realizable as T_{E/F} for fields F containing a primitive pth root of unity or of characteristic p.

We develop a complete set of invariants t_{i}, i = 1, 2, …, p, and u of T - groups as follows. For a pro-p-group Γ, we denote by Γ^{(n)} the nth group in the p-central series of Γ and Γ_{(n)} the nth group in the central series of Γ. Hence Γ^{(1)} = Γ_{(1)} = Γ, and, for n ∈ ℕ, Γ^{(n+1)} = (Γ^{(n)})^{p} · [Γ, Γ^{(n)}] and Γ_{(n+1)} = [Γ, Γ_{(n)}]. For a T -group T we define

t_{1} = dim_{𝔽_{p}} H^{1} (\frac{Z(T)[p]}{Z(T) ∩ T_{(2)}} , 𝔽_{p})
t_{i} = dim_{𝔽_{p}} H^{1} (\frac{Z(T) ∩ T_{(i)}}{Z(T) ∩ T_{(i+1)}} , 𝔽_{p}) , 2 ≤ i ≤ p
u = max{i : 1 ≤ i ≤ p, T^{p} ⊂ T_{(i)}}.

For readability we write t_{i} and u instead of the more cumbersome t_{i}(T) and u(T). Here Z(T) denotes the center of T and Z(T)[p] the subgroup of Z(T) of elements of order dividing p. From our definition of T -groups it follows that T_{(2)} = [T, N] ⊂ N and therefore Z(T) ∩ T_{(2)} ⊂ Z(T)[p], so that t_{1} is well-defined.

Proposition A.1. For arbitrary cardinalities t_{i}, i = 1, 2, …, p, and u with 1 ≤ u ≤ p, the following are equivalent:

(1) The t_{i} and u are invariants of some T -group

(2) (a) if u < p then t_{u} ≥ 1, and (b) if u = p and t_{i} = 0 for all 2 ≤ i ≤ p, then t_{1} ≥ 1.

Moreover, T -groups are uniquely determined up to isomorphism by these invariants.

Remark. In condition (2b) above, one can omit the condition u = p: from (2a) it follows that if u < p and t_{i} = 0 for all 2 ≤ i ≤ p, then u = 1 and therefore t_{1} ≥ 1.

In the following theorem, we consider fields F either containing a primitive pth root of unity or of characteristic p.

Theorem A.2. Suppose p is an odd prime. The following are equivalent:

(1) T is a T -group with invariants t_{i} and u satisfying (a) u ∈ {1, 2},

(b) t_{2} = u − 1, and

(c) t_{i} = 0 for 3 ≤ i < p.

(2) T ≃ T_{E/F} for some cyclic extension of degree p of a field F as above.

Suppose p = 2. Then the following are equivalent:

(1) T is a T -group

(2) T ≃ T_{E/F} for some cyclic extension of degree 2 of a field F.

Corollary. Suppose p is an odd prime, T is a finite nonabelian T - group, N is a maximal closed subgroup of T that is abelian of exponent dividing p, and u is the invariant of T defined above. Set e = dim_{𝔽_{p}} H^{1}(N, 𝔽_{p}) and f = dim_{𝔽_{p}} Z(T). If

e ≢ f + (u − 1) mod p − 1,

then T ≄ T_{E/F} for any cyclic extension E/F of degree p.

Let G be a cyclic group of order p and M_{i}, i = 1, 2, …, p, denote the unique cyclic 𝔽_{p}G-module of dimension i. Since the 𝔽_{p}G-modules we consider will be multiplicative groups, we usually write the action of G exponentially. Recall that for any generator g of G, the element (g − 1) ∈ 𝔽_{p}G is nilpotent of degree p and the fixed 𝔽_{p}G-submodule M_{i}^{G} = M_{i}^{(g−1)^{i−1}} for 1 ≤ i ≤ p. For a set I, let M_{i}^{I} denote the product of |I| copies of M_{i} endowed with the product topology.

We use the word duality exclusively to refer to Pontrjagin duality, between compact and discrete abelian groups and more generally between compact and discrete 𝔽_{p}G-modules. Since we shall consider only the case when G has order p, we do not need to pass to completed group rings. (See, for instance, [Koc, §7.1–7.3 and §14] and [Br].) We denote the Pontrjagin dual of a discrete or compact group Γ by Γ^{∨}.

Lemma A.1. Suppose T is a T -group with invariants t_{i}, i = 1, 2, …, p and u, and N is a maximal closed subgroup of T that is abelian of exponent dividing p. Let σ ∈ T \ N and set σ̄ to be the image of σ in G := T/N. Then

(1) N is a topological 𝔽_{p}G-module.

(2) For any 𝔽_{p}G-submodule M of N and i ≥ 0, ^{i}[T, M] = M^{(σ̄−1)^{i}}.

(3) For i ≥ 2, T_{(i)} = N^{(σ̄−1)^{i−1}}.

(4) There exist sets I_{i}, i = 1, 2, …, p, such that N decomposes into indecomposable 𝔽_{p}G-modules as N = M_{1}^{I_{1}} × M_{2}^{I_{2}} × ⋯ × M_{p}^{I_{p}}, endowed with the product topology, and all such decompositions of N are equivalent. Moreover, for i ≥ 2, t_{i} = |I_{i}|, and

t_{1} = \begin{cases}1 + |I_{1}|, & T is abelian of exponent p \\ |I_{1}|, & otherwise. \end{cases}

(5) T^{p} = ⟨σ^{p}⟩ · T_{(p)}.

(6) If u < p then u is the minimal v with 1 ≤ v ≤ p − 1 such that there is a commutative diagram of pro-p-groups

\xymatrix{1 \ar[r] & N \ar[r] \ar@{->>}[d] & T \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{v} \ar[r] & H \ar[r] & G \ar[r] & 1}

with a lift of σ̄ in H of order p^{2}. If u = p then no such diagram exists for 1 ≤ v ≤ p − 1.

Proof. Since N is a maximal closed subgroup of a pro-p-group T, N is normal of index p. Therefore G is well-defined and is a cyclic group of order p with generator σ̄.

(1). Since N is abelian, the action of T on N factors through G. This action on the pro-p-group N is continuous, whence N is a topological 𝔽_{p}G-module.

(2). Let M be an arbitrary 𝔽_{p}G-submodule of N. Suppose that τ ∈ T is arbitrary, and write τ = nσ^{i} for n ∈ N and i ∈ ℕ ∪ {0}. Since the action of T factors through G we see that [τ, M] = M^{(σ̄^{i}−1)}. Since (σ̄ − 1) divides (σ̄^{i} − 1) for all i ∈ ℕ we obtain that [T, M] = M^{(σ̄−1)}, and the result follows by induction. (Moreover, we observe that if p does not divide i, then [τ, M] = [M, τ] = [T, M] = [M, T].)

(3). Since N is abelian of index p, see that [T, T] = [T, N], and the result follows from (2).

(4). Because 𝔽_{p}G is an Artinian principal ideal ring, every 𝔽_{p}Gmodule U decomposes into a direct sum of cyclic 𝔽_{p}G-modules [SV, Theorem 6.7]. Every cyclic 𝔽_{p}G-module is indecomposable, self-dual, and local. Moreover, each such module has a local endomorphism ring. By the Krull-Schmidt-Azumaya Theorem (see [AF, Theorem 12.6]), all decompositions of an 𝔽_{p}G-module U into indecomposables are equivalent. (In our special case one can check this fact directly.) Applying these results to U = N^{∨} and using duality (see [RZ, Lemma 2.9.4 and Theorem 2.9.6]), we obtain the decomposition and its uniqueness.

Observe that for an 𝔽_{p}G-submodule M of N, Z(T)∩M = M^{G}. Using (3) together with M_{j}^{G} = M_{j}^{(σ̄−1)^{j−1}} for all 1 ≤ j ≤ p, we calculate

Z(T) ∩ N = (M_{1})^{I_{i}} × (M_{2}^{(σ̄−1)})^{I_{2}} × ⋯ × (M_{p}^{(σ̄−1)^{p−1}})^{I_{p}} .

and for i ≥ 2

Z(T) ∩ T_{(i)} = \begin{cases}(M_{i}^{(σ̄−1)^{i−1}})^{I_{i}} × ⋯ × (M_{p}^{(σ̄−1)^{p−1}})^{I_{p}} , & 2 ≤ i ≤ p \\ \{1\}, & i > p. \end{cases}

Then from dim_{𝔽_{p}} M_{j}^{(σ̄−1)^{j−1}} = 1, 1 ≤ j ≤ p, we deduce that t_{i} = |I_{i}|, 2 ≤ i ≤ p.

For the case i = 1, suppose first that T is abelian of exponent p. Then Z(T) = Z(T)[p] = T and T_{(2)} = {1}. By (3), N^{(σ̄−1)} = {1}, from which we deduce |I_{i}| = 0 for i ≥ 2. Therefore

t_{1} = dim_{𝔽_{p}} H^{1}(T, 𝔽_{p}) = dim_{𝔽_{p}} H^{1}(T/N, 𝔽_{p}) + dim_{𝔽_{p}} H^{1}(N, 𝔽_{p})
= 1 + |I_{1}|,

as desired.

Now suppose that T is nonabelian. If an element δ ∈ T \N commutes with N, then T is abelian. Hence no element of T \ N commutes with N and thus Z(T) ⊂ N. We obtain Z(T)[p] = Z(T) ∩ N and so

t_{1} = dim_{𝔽_{p}} H^{1} (\frac{Z(T) ∩ N}{Z(T) ∩ T_{(2)}} , 𝔽_{p}) = |I_{1}|.

Finally, assume that T is abelian and not of exponent p. Then the endomorphism ϕ : x ↦ x^{p} satisfies im ϕ ≠ {1} and N ⊂ ker ϕ. Since N is a maximal closed subgroup, we obtain ker ϕ = N, that is, N = T [p]. Since Z(T) = T, we deduce t_{1} = dim_{𝔽_{p}} H^{1}(N, 𝔽_{p}) = |I_{1}|.

(5). For δ ∈ N we have (δσ)^{2} = [σ, δ]δ^{2}σ^{2} and, by induction,

(δσ)^{i} = [\underbrace{σ, [σ, …, [σ}_{i−1 times}, δ] ⋯]]^{(\begin{matrix}i \\ i \end{matrix})} ⋯ [σ, [σ, δ]]^{(\begin{matrix}i \\ 3 \end{matrix})}[σ, δ]^{(\begin{matrix}i \\ 2 \end{matrix})}δ^{i}σ^{i}.

Then (Nσ)^{p} = (σ^{p}) · ^{p−1}[σ, N], which by (2) and (3) may be written σ^{p}·T_{(p)}. Replacing σ with σ^{v} for (v, p) = 1, we conclude T^{p} = ⟨σ^{p}⟩·T_{(p)}.

(6). Suppose that for some v < u there is a commutative diagram

\xymatrix{1 \ar[r] & N \ar[r] \ar@{->>}[d] & T \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{v} \ar[r] & H \ar[r] & G \ar[r] & 1}

with a lift of σ̄ in H of order p^{2}. Then the surjection T ↠ H factors through T/N^{(σ̄−1)^{v}}. But by (3), T_{(v+1)} = N^{(σ̄−1)^{v}} and by definition of u, we have T^{p} ⊂ T_{(u)} ⊂ T_{(v+1)}. Hence every lift of σ̄ into T/N^{(σ̄−1)^{v}} is of order p, and the same holds for H. We conclude that no commutative diagram as above with σ̄ lifting to an element of order p^{2} exists for v < u.

Going the other direction, suppose that u < p and consider σ^{p}. By (5), T^{p} = ⟨σ^{p}⟩ · T_{(p)}, and T_{(p)} ⊂ T_{(u+1)}. By definition of u, we have T^{p} ⊂ T_{(u)} and T^{p} ⊄ T_{(u+1)}. We conclude that σ^{p} ∈ T_{(u)} \ T_{(u+1)}. Now σ^{p} ∈ N since N is a maximal closed subgroup. By (3), since T_{(u)} = N^{(σ−1)^{u−1}} for u ≥ 2, we obtain σ^{p} ∈ N^{(σ−1)^{u−1}} \ N^{(σ−1)^{u}}. From [σ, σ^{p}] = 1 we obtain σ^{p} ∈ N^{G}. Therefore

σ^{p} ∈ N^{G} ∩ N^{(σ−1)^{u−1}} \ N^{G} ∩ N^{(σ−1)^{u}}.

We claim that there exists an 𝔽_{p}G-submodule M_{u} of N such that M_{u}^{G} = ⟨σ^{p}⟩ and N = M_{u}×Ñ for some 𝔽_{p}G-submodule Ñ of N. Assume a factorization of N into cyclic 𝔽_{p}G-submodules as in (4). Let w ∈ N such that w^{(σ−1)^{u−1}} = σ^{p}. Consider the components of w in the factors of N. Without loss of generality the nonzero components of w lie in factors of dimension at least u, and there must exist at least one such factor M_{u} of dimension precisely u such that the projection proj_{M_{u}} w of w generates M_{u} as an 𝔽_{p}G-module. Let Ñ be the 𝔽_{p}G-submodule of N generated by all of the factors of N except M_{u}. Then it follows that N = M_{u} × Ñ. Factoring T by Ñ, we obtain the commutative diagram

\xymatrix{1 \ar[r] & N \ar[r] \ar@{->>}[d] & T \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{u} \ar[r] & T/Ñ \ar[r] & G \ar[r] & 1}

in which a lift of σ̄ is of order p^{2}.

We have shown that if u < p, then u is the minimal v in 1 ≤ v ≤ p−1 such that a commutative diagram

\xymatrix{1 \ar[r] & N \ar[r] \ar@{->>}[d] & T \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{v} \ar[r] & H \ar[r] & G \ar[r] & 1}

exists with a lift of σ̄ of order p^{2}, and if u = p then no such commutative diagram with a lift of σ̄ of order p^{2} exists for 1 ≤ v ≤ p − 1. □

Remark. Observe that from Lemma A.1(3,4) it follows that the nilpotent index of T is at most p. Moreover, this index is ≤ p−1 if and only if I_{p} = ∅. In this case, each finite quotient of T is a regular p-group. (See [Ha, §12.4].) This fact can also be seen from the identity expressing (σδ)^{p} as a product of pth-powers and commutators as in the proof of (5) above.

Lemma A.2. Let T be a T -group with invariants t_{i}, i = 1, 2, …, p, and u.

(1) T is abelian if and only if t_{i} = 0 for all 2 ≤ i ≤ p.

(2) T is of exponent p if and only if u = p and t_{u} = 0.

(3) If u < p then t_{u} ≥ 1.

(4) If t_{i} = 0 for all 2 ≤ i ≤ p, then t_{1} ≥ 1.

Proof. (1). If T is abelian then T_{(2)} = {1} and therefore t_{i} = 0 for i ≥ 2. Conversely, suppose that t_{i} = 0 for i ≥ 2. Then by Lemma A.1(4), we have N^{G} = N. By Lemma A.1(3), T_{(2)} = {1}, whence T is abelian.

(2). If u = p and t_{p} = 0 then the fact that T is of exponent p follows from the definitions. We show the converse, as follows. Assume that T is of exponent p. From Lemma A.1(5) we have T_{(p)} = T^{p} = {1}. Hence t_{p} = 0, and the definition of u gives u = p.

(3). Suppose that u < p. Since T^{p} ⊂ T_{(u)} and T^{p} ⊄ T_{(u+1)}, we cannot have T^{p} = {1}. Let δ ∈ T be arbitrary with δ^{p} ≠ 1. Then δ ∉ N, and since N is of index p, δ^{p} ∈ N. Since [δ, δ^{p}] = 1 we deduce δ^{p} ∈ Z(T)∩N. We have obtained T^{p} ⊂ Z(T) ∩ T_{(u)} and T^{p} ⊄ Z(T) ∩ T_{(u+1)}, whence t_{u} ≥ 1.

(4). Suppose t_{i} = 0 for all 2 ≤ i ≤ p. From (1) we conclude that T is abelian. Hence from the definition of t_{1} and the assumption that T is nontrivial, we conclude that t_{1} ≥ 1. □

Remark. Observe that if p = 2 then (2) implies (1) in Lemma A.2, which is good because every group of exponent 2 is abelian!

Proof of Proposition A.1. By Lemma A.2(3,4), we see that conditions (2) on the t_{i} and u hold for any T -group. Hence (1) implies (2). We now show that (2) implies (1). Suppose we are given cardinalities t_{i}, i = 1, 2, …, p, and u satisfying conditions (2).

The case u < p. Let G be a group of order p and I_{i}, i = 1, ⋯, p sets with cardinalities |I_{i}| satisfying |I_{i}| = t_{i} for i ≠ u and 1 + |I_{u}| = t_{u}. Set N = X × M_{1}^{I_{1}} × M_{2}^{I_{2}} × ⋯ × M_{p}^{I_{p}}, where X ≃ M_{u} and N is endowed with the product topology. Observe that N possesses a system of neighborhoods of the identity consisting of open G-invariant normal subgroups. Hence N is a pseudocompact 𝔽_{p}G-module. (See [Br, page 443].)

Define an action of ℤ_{p} on N by letting a generator σ of ℤ_{p} act via a generator of G. (We write ℤ_{p} multiplicatively to reduce confusion.) We see that ℤ_{p} acts continuously on N and we form the semidirect product N ⋊ ℤ_{p} in the category of pro-p-groups. Now choose an 𝔽_{p}G-module generator x of X and define x_{i} = x^{(σ−1)^{i}} for 0 ≤ i ≤ u. Since (σ − 1) is nilpotent of degree u on X we obtain x_{u−1} ≠ 1 and x_{u} = 1. We set R to be the closed subgroup ⟨σ^{p}x_{u−1}⟩ ⊂ N ⋊ ℤ_{p}. Observe that R is normal since x_{u} = 1. Finally we form Γ = (N ⋊ ℤ_{p})/R and set Δ to be the image of N ⋊ {1} in Γ. Since Δ ≃ N as pro-p-G operator groups, we identify them in what follows.

By construction Δ is a maximal closed subgroup of Γ which is abelian of exponent p. Hence Γ is a T -group. Since the image of σ in Γ has order p^{2} we see that Γ is not of exponent p. From the decomposition of N, we obtain by Lemma A.1(4) that the invariants t_{i} are as desired. It remains only to show that u is as given. By Lemma A.1(5) we have Γ^{p} = ⟨x_{u−1}⟩ · Γ_{(p)}. From Lemma A.1(2,3) we calculate that x_{u−1} ∈ Γ_{(u)} and x_{u−1} ∉ Γ_{(u+1)}. Hence u is as desired.

The case u = p. We proceed analogously. Let G be a group of order p and I_{i}, i = 1, 2, …, p sets with cardinalities |I_{i}| satisfying

|I_{i}| = t_{i},  2 ≤ i ≤ p
|I_{i}| = t_{i},  i = 1 and some t_{j} ≠ 0, 2 ≤ j ≤ p
1 + |I_{i}| = t_{i},  i = 1 and t_{j} = 0, 2 ≤ j ≤ p.

Set N = M_{1}^{I_{1}} ×M_{2}^{I_{2}} ×⋯×M_{p}^{I_{p}}, where N is endowed with the product topology. Set Γ = N ⋊ G, Δ = N ⋊ {1}, and let σ be a generator of G.

By construction Δ is a maximal closed subgroup of Γ which is abelian of exponent p. Hence Γ is a T -group and is abelian if and only if t_{i} = 0 for i ≥ 2, and in this case Γ is of exponent p. From the decomposition of N we obtain by Lemma A.1(4) that the invariants t_{i} are as desired. It remains only to show that u is as given. By Lemma A.1(5) we have Γ^{p} = Γ_{(p)} and u = p, as desired.

Now we show that T -groups are uniquely determined up to isomorphism by the invariants t_{i} and u. Let T be an arbitrary T -group with invariants t_{i}, i = 1, 2, …, p, and u, and N be a maximal closed subgroup of T that is abelian of exponent dividing p. Let G = T/N. From Lemma A.1(4) the structure of N as an 𝔽_{p}G-module is determined up to isomorphism, and T is an extension of N by G. Let σ ∈ T \N. Since N is of index p we have that σ^{p} ∈ N, and since σ̄ ∈ G generates G we have that σ^{p} ∈ N^{G}. It remains only to determine the isomorphism class of N as an 𝔽_{p}G-module with a distinguished factor X such that σ^{p} ∈ X^{G}.

Case 1 : u = p, t_{1} ≥ 1, and t_{i} = 0 for all 2 ≤ i ≤ p. From Lemma A.2(1) we see that T is abelian so that T_{(2)} = {1}. Then from the definition of u we see that T has exponent p. We deduce that T ≃ M_{1}^{I_{1}} × G where G is a cyclic group of order p and t_{1} = |I_{1}| + 1. Thus T is determined by the invariants.

Case 2 : t_{1} ≥ 1, t_{i} = 0 for all 2 ≤ i ≤ p, and u = 1. Again we deduce that T is abelian. From the definition of u we see that T does not have exponent p, and so by the definition of T -group, T has exponent p^{2}. We deduce that N = X × Ñ where σ^{p} generates an 𝔽_{p}G-module X isomorphic to M_{1} and Ñ ≃ M_{1}^{I′} with |I′| + 1 = t_{1}. Thus T is determined by the invariants.

Case 3 : t_{i} ≠ 0 for some i with 2 ≤ i ≤ p. By Lemma A.2(1) we see that T is nonabelian. By Lemma A.1(5), T^{p} = ⟨σ^{p}⟩ · T_{(p)}. From Lemma A.1(3) and the definition of u, we deduce that σ^{p} ∈ N^{(σ−1)^{u−1}} and, if u < p, that σ^{p} ∉ N^{(σ−1)^{u}}. When u < p we obtain

σ^{p} ∈ (N^{G} ∩ N^{(σ−1)^{u−1}}) \ (N^{G} ∩ N^{(σ−1)^{u}}),

while when u = p, we have σ^{p} ∈ N^{(σ−1)^{p−1}}.

If u < p then by Proposition A.1 we have t_{u} ≥ 1, and using the same argument as in the proof of Lemma A.1(6), we see that N contains a distinguished direct factor X ≃ M_{u} such that X^{G} = ⟨σ^{p}⟩. We deduce that N = X × Ñ for Ñ ≃ M_{u}^{I′} × ∏_{i≠u} M_{i}^{I_{i}} for sets I_{i}, i ≠ u, and I′ such that |I_{i}| = t_{i}, i ≠ u, and 1 + |I′| = t_{u}. Then T is determined by its invariants.

If u = p, we claim that without loss of generality we may assume that σ^{p} = 1. Since σ^{p} ∈ N^{(σ−1)^{p−1}}, let ν ∈ N such that σ^{p} = ν^{(σ−1)^{p−1}} and set τ = σν^{−1}. Then τ ∈ T \ N and τ^{p} = σ^{p}(ν^{−1})^{1+σ+···+σ^{p−1}}. Since (σ − 1)^{p−1} = 1 + σ + ⋯ + σ^{p−1} in 𝔽_{p}G we deduce that τ^{p} = 1, as desired. Hence T = N ⋊ G and is determined by its invariants. □

Lemma A.3. Suppose that Γ is a profinite group such that Γ(p) is a free pro-p-group of (possibly infinite) rank n, and let Δ be a normal subgroup of Γ of index p. Then the invariants of the T -group Γ/Δ^{p}[Δ, Δ] are given by

t_{i} = \begin{cases}1, & i = 1 \\ 0, & 2 ≤ i < p \\ n − 1, & i = p, n < ∞ \\ n, & i = p, n an infinite cardinal \end{cases}

u = 1.

We observe the convention that a free pro-p-group is a free pro-p-group on a positive number of generators.

Proof. Since Γ/Δ^{p}[Δ, Δ] = Γ(p)/Φ(Δ(p)), we may assume without loss of generality that Γ is a free pro-p-group. Let {σ_{i}}_{i∈I}∪{σ} be a minimal generating set of Γ such that σ ∉ Δ and all σ_{i} ∈ Δ. Write σ̄_{i} for the image of σ_{i} in T := Γ/Δ^{(2)}. We use the analogue of the Kurosh subgroup theorem in the context of pro-p-groups.

For each i ∈ I, let Q_{i} be the closed subgroup of Γ generated by σ_{i}, and let Q be the closed subgroup generated by σ. Then Γ is the free product (⋆_{i∈I} Q_{i})⋆Q in the category of pro-p-groups. Applying [NSW, Theorem 4.2.1] we see that

Δ = (⋆_{0≤i,j≤p−1} Q_{i}^{σ^{j}}) ⋆ (Q^{p} ∩ Δ),

where Q_{i}^{σ^{j}} = σ^{−j}Q_{i}σ^{j} and Q^{p} is the closed subgroup of Q generated by σ^{p}. We deduce that Δ/Δ^{(2)} is a topological product of copies of ℤ/pℤ corresponding to the generators of the factors of Δ above.

For each i ∈ I we obtain the 𝔽_{p}G-submodule of Δ

Y_{i} := ⟨σ̄_{i}, σ̄^{−1}σ̄_{i}σ̄, …, σ̄^{−(p−1)}σ̄_{i}σ̄^{p−1}⟩ = ⟨σ̄_{i}, σ̄_{i}^{σ̄−1}, …, σ_{i}^{(σ̄−1)^{p−1}}⟩ ≃ M_{p},

where the equality follows from the fact that Y_{i} is the cyclic 𝔽_{p}Gmodule generated by σ̄_{i}. We deduce that Δ/Δ^{(2)} ≃ M_{1} × M_{p}^{I_{p}}, where |I_{p}| + 1 = n. We calculate that u = 1 since σ^{p} ∈ M_{1} and σ^{p} ∉ T_{(2)}. Finally Lemma A.1(4) tells us that t_{i}, i = 1, 2, …, p, have the prescribed values. □

Lemma A.4. Let S be a free pro-p-group. Then there exists a field F of characteristic 0 such that G_{F} ≃ S.

Proof. First we let F_{0} be any algebraically closed field of characteristic 0 with cardinality greater than or equal to d = dim_{𝔽_{p}} H^{1}(S, 𝔽_{p}); such a field exists as each field admits an algebraic closure. Set F_{1} := F_{0}(t). By [Do] (see also [vDR]), G_{F_{1}} is a free profinite group, and we let P denote a p-Sylow subgroup of G_{F_{1}}. By [RZ, Corollary 7.7.6], P is a free pro-p-group. Let F_{2} be the fixed field of P.

Consider the set A of linear polynomials t − c ∈ F_{1} for c ∈ F_{0}. Because F_{0}[t] is a unique factorization domain we obtain that the classes of these polynomials in F_{1}^{×}/F_{1}^{×p} are linearly independent over 𝔽_{p}. Choose a subset of cardinality d of A, and let V be the vector subspace of F_{1}^{×}/F_{1}^{×p} generated by the classes of the elements of A. Thus dim_{𝔽_{p}} V = d.

Since [F_{2} : F_{1}] is coprime to p (for results on supernatural numbers, see [S, Chapter 1]), we deduce that V injects into F_{2}^{×}/F_{2}^{×p}. Let W denote this image. Now let F be a maximal algebraic field extension of F_{2} such that W injects into F^{×}/F^{×p}.

We claim that the image i(W) of W in F^{×}/F^{×p} is in fact F^{×}/F^{×p}. Suppose not. Then there exists a class [f] ∈ F^{×}/F^{×p}, f ∈ F^{×}, such that [f] ∉ i(W). Then the subgroup ⟨[f]⟩ is of order p with trivial intersection with i(W). Set L = F (\sqrt[p]{f}). We have that W injects into L^{×}/L^{×p}, but L is strictly larger than F, a contradiction. The rank of G_{F} is then dim_{𝔽_{p}} H^{1}(G_{F} , 𝔽_{p}) = dim_{𝔽_{p}} F^{×}/F^{×p} = d. □

Proof of Theorem A.2. The case p = 2. Let u ∈ {1, 2}, t_{1}, and t_{2} be invariants of a T -group T. By Proposition A.1, we have that t_{1} ≥ 1 if u = 1, while if u = 2 then either t_{1} ≥ 1 or t_{2} ≥ 1.

Case 1 : T is not of exponent 2. By Lemma A.2(2) we find that either u = 1 or t_{2} ≥ 1. Let G be a group of order 2 and N = M_{1}^{I_{1}} × M_{2}^{I_{2}} a topological 𝔽_{2}G-module for some sets I_{1} and I_{2} satisfying |I_{1}| = t_{1} and |I_{2}| = t_{2}, and let M = N^{∨}. From [MSw, Corollary 2], there exists a field extension E/F of fields of characteristic not 2 with Gal(E/F) ≃ G such that H^{1}(G_{E}, 𝔽_{2}) ≃ E^{×}/E^{×2} ≃ M as 𝔽_{2}G-modules if and only if there exist Υ ∈ {0, 1} and cardinalities d and e such that t_{1}+1 = 2Υ+d; t_{2} + Υ = e; if Υ = 0 then d ≥ 1; and if Υ = 1 then e ≥ 1. Moreover, the resulting extension E/F will have −1 ∈ N_{E/F} (E^{×}) if and only if Υ = 1. Finally, by [MSw, proof of Theorem 1], we may choose E/F such that G_{F} is a pro-2-group.

If u = 1 then we set Υ = 1 and e = t_{2} + Υ. Since t_{1} ≥ 1 we may choose d ≥ 0 such that 2Υ + d = t_{1} + 1. We have e ≥ 1 and hence we have satisfied the conditions for a field extension E/F with G_{E}/G_{E}^{(2)} ≃ M^{∨} ≃ N. Since Υ = 1 we have −1 ∈ N_{E/F} (E^{×}) and by Albert’s criterion [A], E/F embeds in a cyclic extension E′/F of degree 4. We see that E′/E is degree 2 and hence Gal(E′/E) is a quotient of G_{E}/G_{E}^{(2)}. Let Gal(E/F) = ⟨σ̄⟩. Then we have the commutative diagram

\xymatrix{1 \ar[r] & G_{E}/G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G_{F} /G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{1} \ar[r] & H \ar[r] & G \ar[r] & 1}

in which σ̄ lifts to an element of order 4. We deduce from Lemma A.1(6) that the invariant u for T_{E/F} is 1. By Lemma A.2(2), T_{E/F} is not of exponent 2. From Lemma A.1(4) and the isomorphism G_{E}/G_{E}^{(2)} ≃ N ≃ M_{1}^{I_{1}} × M_{2}^{I_{2}} we find that the invariants t_{1} and t_{2} of T_{E/F} match those of T. Hence the T -groups T and T_{E/F} have the same invariants and we conclude by Proposition A.1 that T ≃ T_{E/F}.

If u = 2 then we determined that t_{2} ≥ 1. We take Υ = 0, d = t_{1}+1 ≥ 1, and e = t_{2} and obtain a cyclic extension E/F as before. Since Υ = 0 we have −1 ∉ N_{E/F} (E^{×}) and by Albert’s criterion [A], E/F does not embed in a cyclic extension E′/F of degree 4. Let Gal(E/F) = ⟨σ̄⟩. We have that there is no commutative diagram

\xymatrix{1 \ar[r] & G_{E}/G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G_{F} /G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{1} \ar[r] & H \ar[r] & G \ar[r] & 1}

in which σ̄ lifts to an element of order 4. We deduce from Lemma A.1(6) that the invariant u of T_{E/F} is 2. Because t_{2} ≥ 1, G_{E}/G_{E}^{(2)} contains an 𝔽_{2}G-submodule isomorphic to M_{2}, and σ̄ does not act trivially on M_{2}. Hence T_{E/F} is nonabelian. By Lemma A.1(4) and the isomorphism G_{E}/G_{E}^{(2)} ≃ N ≃ M_{1}^{I_{1}} × M_{2}^{I_{2}} we deduce that the invariants t_{1} and t_{2} of T_{E/F} match those of T. Hence the T -groups T and T_{E/F} have the same invariants and we conclude by Proposition A.1 that T ≃ T_{E/F}.

Case 2 : T has exponent 2. By Lemma A.2(2) we have that u = 2 and t_{2} = 0, and therefore by the restrictions for abstract T -groups we deduce t_{1} ≥ 1. Let N = M_{1}^{I_{1}} be a topological 𝔽_{2}G-module for a set I_{1} satisfying |I_{1}|+1 = t_{1}. Choose Υ = 0, d = t_{1}+1, and e = t_{2} = 0. Then d ≥ 1 and there exists a field extension E/F of fields of characteristic not 2 such that H^{1}(G_{E}, 𝔽_{2}) ≃ M. Moreover −1 ∉ N_{E/F} (E^{×}) if and only if Υ = 0. As before we deduce that u = 2 for T_{E/F}, and by Lemma A.1(4) we find that t_{2} of T_{E/F} is 0. Hence by Lemma A.2(2), T_{E/F} has exponent 2. By Lemma A.1(4) again we obtain that t_{1} for T_{E/F} is equal to t_{1}. Hence the T -groups T and T_{E/F} have the same invariants and we conclude by Proposition A.1 that T ≃ T_{E/F}.

We have shown then that for p = 2, any abstract T -group may be realized as the T -group of E/F for a cyclic extension of degree 2 of fields of characteristic not 2. Hence the equivalence holds for all E/F.

The case p > 2. First we characterize those T -groups occurring as T_{E/F} for fields F such that the maximal pro-p-quotient G_{F} (p) of the absolute Galois group G_{F} is free pro-p. Lemma A.3 tells us that for such a field F and a cyclic extension E/F of degree p, the invariants of T_{E/F} must satisfy t_{1} = 1, t_{i} = 0 for 2 ≤ i < p, and u = 1, and that the rank of G_{F} (p) is one more than the invariant t_{p}. Now suppose that T is a T -group with invariants t_{1} = 1, t_{i} = 0 for 2 ≤ i < p, and u = 1. By Lemma A.4 there exists a field F such that G_{F} is a free pro-p-group of rank t_{p} + 1. Letting Δ be any maximal closed subgroup of G_{F} and E = Fix(Δ), Lemma A.3 tells us that the invariants of T_{E/F} match those of T, and then by Proposition A.1 we have that T ≃ T_{E/F}. Therefore the T -groups which occur as T_{E/F} for fields F with free maximal pro-p-quotient G_{F} (p) are precisely those for which t_{1} = 1, t_{i} = 0 for 2 ≤ i < p, and u = 1.

Now we characterize which of the remaining T -groups occur as T_{E/F} for cyclic field extensions E/F of degree p for F a field either of characteristic p or containing a primitive pth root of unity. Because the remaining T -groups cannot be T_{E/F} for a cyclic field extension E of a field F with G_{F} (p) a free pro-p maximal pro-p-quotient G_{F} (p), by Witt’s Theorem we have that the characteristic of F is not p. Hence we consider only fields F of characteristic not p and containing a primitive pth root of unity.

For a cyclic extension E/F of degree p, consider the group

M_{E/F} := H^{1}(G_{E}, 𝔽_{p}) ≃ E^{×}/E^{×p}.

This group M_{E/F} is an 𝔽_{p} Gal(E/F)-module. Now let G be an abstract group of order p. Since the particular isomorphism G ≃ Gal(E/F) does not alter the isomorphism class of M_{E/F} as an 𝔽_{p}G-module, we may consider all such modules 𝔽_{p}G-modules. (See [MSw].)

By [MSw, Corollary 1], M ≃ M_{E/F} as 𝔽_{p}G-modules for suitable E/F with G ≃ Gal(E/F) if and only if

M = M_{1}^{I_{1}} ⊕ M_{2}^{I_{2}} ⊕ M_{p}^{I_{p}}

where the cardinalities j_{1} = |I_{1}|, j_{2} = |I_{2}|, and j_{p} = |I_{p}| satisfy the following conditions: j_{1} + 1 = 2Υ + d, j_{2} = 1 − Υ, and j_{p} + 1 = e for some cardinalities d, e and Υ ∈ {0, 1} where d ≥ 1 if Υ = 0 and e ≥ 1. (Recall that further, by the Krull-Schmidt-Azumaya theorem, the decomposition above into indecomposable 𝔽_{p}G-modules is unique up to equivalence.) Moreover, in the given E/F we have Υ = 1 if and only if ξ_{p} ∈ N_{E/F} (E^{×}). Finally, by [MSw, proof of Theorem 1], we may choose E/F such that G_{F} is a pro-p-group.

We observe that the constraints on j_{1}, j_{2}, and j_{p} are then j_{1} ≥ Υ and j_{2} = 1 − Υ. Now by duality, H^{1}(G_{E}, 𝔽_{p})^{∨} ≃ G_{E}/G_{E}^{(2)}, and since cyclic 𝔽_{p}G-modules are self-dual, we may derive conditions on the topological 𝔽_{p}G-module G_{E}/G_{E}^{(2)} occurring as maximal closed subgroups of T -groups G_{F} /G_{E}^{(2)}, as follows. Set G = G_{F} /G_{E}.

First we relate Υ and the invariant u. We claim that for any T -group G_{F} /G_{E}^{(2)}, we have u ≤ 2. Write E = F (\sqrt[p]{a}) for some a ∈ F^{×}. Let [e] ∈ E^{×}/E^{×p} denote the class of e ∈ E^{×}. Then X = ⟨[\sqrt[p]{a}], [ξ_{p}]⟩ is a cyclic 𝔽_{p}G-submodule of M and is isomorphic to M_{i} for some i ∈ \{1, 2\}. By equivariant Kummer theory (see [W]), L = E( \sqrt[p^{2}]{a}, ξ_{p^{2}}) is a Galois extension of F. Moreover, G_{F} /G_{L} is a homomorphic image of G_{F} /G_{E}^{(2)}, since L/E is an elementary abelian extension. Then L/F (ξ_{p^{2}}) is Galois with group ℤ/p^{2}ℤ, and for any σ ∈ G_{F} \ G_{E}, the restriction σ_{L} ∈ Gal(L/F) restricts to a generator. Hence 1 ≠ σ^{p} ∈ Gal(L/E), and therefore 1 ≠ σ^{p} ∈ G_{E}/G_{E}^{(2)}. We have a commutative diagram

\xymatrix{1 \ar[r] & G_{E}/G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G_{F} /G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{i} \ar[r] & G_{F} /G_{L} \ar[r] & G \ar[r] & 1}

and so by Lemma A.1(6) we deduce that u ≤ i ≤ 2.

Now we claim that Υ = 1 if and only if u = 1. We have that Υ = 1 if and only if ξ_{p} ∈ N_{E/F} (E^{×}). By Albert’s criterion [A], Υ = 1 if and only if E/F embeds in a cyclic extension of F of degree p^{2}, if and only if there exists a closed normal subgroup Ñ ⊂ G_{E} such that G_{F} /Ñ ≃ ℤ/p^{2}ℤ. Any such closed normal subgroup must contain G_{E}^{(2)}. Hence we deduce that Υ = 1 if and only if there exists a commutative diagram

\xymatrix{1 \ar[r] & G_{E}/G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G_{F} /G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{1} \ar[r] & H \ar[r] & G \ar[r] & 1}

in which nontrivial elements of G lift to elements of order p^{2}. By Lemma A.1(6), we deduce that Υ = 1 if and only if u = 1.

We have therefore shown that u ≤ 2 and u = 2 − Υ. Translating the remaining conditions on j_{1} and j_{2}, we see that j_{1} ≥ 2 − u and j_{2} = u − 1. Now by Lemma A.1(4), t_{2} = j_{2}, t_{i} = 0 for 3 ≤ i ≤ p − 1, and t_{p} = j_{p}. Moreover, t_{1} = j_{1} if T is not abelian of exponent p. By Lemma A.2(2), T is of exponent p if and only if u = p. But we have shown that u ≤ 2 < p, whence T is not of exponent p and we have t_{1} = j_{1}. By Proposition A.1, if u = 1 then t_{1} ≥ 1, and since t_{2} = u − 1 the conditions for applying [MSw, Corollary 1] are valid. We conclude that a T -group T_{E/F} with prescribed invariants subject to conditions (1) exists. □

Lemma A.5. Let T be a T -group with invariants t_{i}, i = 1, 2, …, p, and u, and let N be a maximal closed subgroup of T which is abelian of exponent dividing p. Then

1 + dim_{𝔽_{p}} H^{1}(N, 𝔽_{p}) =

\begin{cases}1 + ∑_{i=1}^{p} it_{i}, & u < p or t_{i} > 0 for some 2 ≤ i ≤ p. \\ t_{1} & otherwise. \end{cases}

Furthermore, if t_{i} > 0 for some 2 ≤ i ≤ p or if u = p and t_{i} = 0 for 2 ≤ i ≤ p, then Z(T) has exponent p and dim_{𝔽_{p}} H^{1}(Z(T), 𝔽_{p}) = ∑_{i=1}^{p} t_{i}.

Proof. Both statements follow from Lemma A.2(1,2) and Lemma A.1(4). □

Proof of Corollary to Theorem A.2. Assume that T ≃ T_{E/F} for some E/F of degree p. Theorem A.2 gives us that u ∈ {1, 2}, so u < p. By duality e = dim_{𝔽_{p}} N/N^{(2)} = log_{p} |N|, and log_{p} |T | = 1 + e since N is of index p. By Lemma A.5 we have e = ∑_{i=1}^{p} it_{i}. Since T is nonabelian, by Lemma A.2(1) we have that some t_{i}, 2 ≤ i ≤ p, is nonzero. Moreover, from the facts that N is of index p, T is nonabelian, and N is abelian of exponent p, we deduce that Z(T) ⊂ N. Hence Z(T) is an 𝔽_{p}- vector space and dim_{𝔽_{p}} Z(T) is well-defined. By Lemma A.5 we have f = ∑_{i=1}^{p} t_{i}.

If u = 1 then by Theorem A.2 we deduce t_{i} = 0, 2 ≤ i < p, and therefore e ≡ f mod (p − 1), a contradiction. If u = 2 then by Theorem A.2 we deduce t_{2} = 1 and t_{i} = 0, 3 ≤ i < p. Therefore e ≡ f +1 mod (p−1), a contradiction. □

Now we consider to what extent the maximal closed subgroup N in the definition of a T -group is unique. For p > 2, let H denote the Heisenberg group of order p^{3}, and for p = 2, let H denote the dihedral group of order 8. In both cases we have the presentation

H = ⟨a, b | a^{p} = b^{p} = [a, b]^{p} = 1, [a, b] is central in H⟩

Observe that N_{a} = ⟨a, [a, b]⟩ and N_{b} = ⟨b, [a, b]⟩ are distinct maximal subgroups which are abelian of exponent p. Now let T = H ×(ℤ/pℤ)^{I}, endowed with the product topology, for any index set I. Then N_{a} × (ℤ/pℤ)^{I} and N_{b}×(ℤ/pℤ)^{I} are again distinct maximal closed subgroups which are abelian of exponent p. We show these groups, together with the elementary abelian T -groups of order greater than p, are the only T - groups with more than one maximal closed subgroup which is abelian of exponent dividing p. Outside of these cases, then, the N in the definition of a T -group is a characteristic subgroup.

Proposition A.2. Let T be a T -group. The following are equivalent:

• T has at least two maximal closed subgroups which are abelian of exponent dividing p

• T is either elementary abelian of order greater than p, or the direct product of H and an abelian group of exponent dividing p.

Proof. Let T be a T -group and suppose that N and M are distinct closed index p subgroups of exponent dividing p. Then N and M are normal, S = M ∩ N is a normal closed subgroup of T of index p^{2}, and M/S and N/S are distinct subgroups of T/S of order p with trivial intersection. There exists an element σS ∈ T/S contained in neither M/S nor N/S. The actions of T/N on N and T/M on M are then both given by conjugation by σ. We then identify T/N and T/M as G = ⟨σN⟩ = ⟨σM⟩.

Consider the image of M under the natural homomorphism T → G = T/N. Since M is not contained in N, we see that the image of M is G. Hence M contains an element m = σn for some element n ∈ N. Since M is abelian, m = σn acts trivially on M. Hence G acts trivially on M ∩ N. By Lemma A.1(3), T_{(2)} = M^{(σ−1)} = N^{(σ−1)}, whence N^{(σ−1)} ⊂ M ∩N. Because G acts trivially on N^{(σ−1)}, we obtain that N^{(σ−1)^{2}} = {0}.

By Lemma A.1(4), N = M_{1}^{I_{1}} × ⋯ × M_{p}^{I_{p}} for some sets I_{i}, and the invariants t_{i}, i ≥ 2, of T satisfy t_{i} = |I_{i}|. If p > 2, then from N^{(σ−1)^{2}} = {0} we deduce t_{i} = 0, i ≥ 3. Therefore N = M_{1}^{I_{1}} × M_{2}^{I_{2}} for sets I_{1}, I_{2}.

We now show that the invariant u of T is equal to p. Assume first that p > 2. By the proof of Lemma A.1(5), 1 = m^{p} = (σn)^{p} = σ^{p}v for some v ∈ T_{(p)}. By Lemma A.1(3), T_{(p)} ⊂ M_{p}^{I_{p}}, while we have already shown that t_{p} = |I_{p}| = 0. Lemma A.1(5) then gives us that T is of exponent p, and so by Lemma A.2, u = p, as desired. Now assume that p = 2 and u = 1. By Lemma A.1(5), T^{2} = ⟨σ^{2}⟩T_{(2)}. From u = 1 we see that σ^{2} ∉ T_{(2)}. Again by the proof of Lemma A.1(5), m^{2} = (σn)^{2} = σ^{2}v for some v ∈ T_{(2)}. But then m^{2} ≠ 1, a contradiction. Hence u = 2 in the case p = 2 as well.

Case A: N ⊄ Z(T). Since G acts trivially on M ∩ N and M ∩ N is an index p subgroup of N, we obtain that t_{2} = |I_{2}| = 1. Hence the invariants of T satisfy t_{2} = 1, t_{i} = 0 for i ≥ 3, and u = p. Since the T -group U = (M_{1}^{I_{1}} × M_{2}) ⋊ G = M_{1}^{I_{1}} × H has the same invariants, by Proposition A.1, T ≃ U.

Case B: N ⊂ Z(T). Since N is index p, we obtain that T is abelian. Since N is of index p, T must be either be itself of exponent p or the direct product of ℤ/p^{2}ℤ and (ℤ/pℤ)^{I} for some index set I. In this latter case, observe that no element of order p^{2} can be contained in N. Hence every element of N lies in W = pℤ/p^{2}ℤ × (ℤ/pℤ)^{I}. But W is of index p in T and so N = M = W, a contradiction. Hence T is abelian of exponent p, and since M and N are distinct subgroups of index p, |T | > p. □

Then Theorem A.2 and the determination of the invariants u in the proof of the previous proposition give the following

Corollary A.3. Suppose p > 2. Then each T_{E/F} contains precisely one maximal closed subgroup which is elementary abelian.

2. Proof of Theorem A.1

Proof of Theorem A.1. Suppose Γ is a pro-p-group with maximal closed subgroup Δ, and let T = Γ/Δ^{(2)}, N = Δ/Δ^{(2)} and G = Γ/Δ. Write σ̄ and τ̄ for the images of σ and τ, respectively.

(1). Since Δ surjects onto N we have that σ̄ ∉ N, τ̄ ∈ N, and

e[σ̄, τ̄] ∉ ^{p−1}[σ̄, N]
e+1[σ̄, τ̄] = 1.

By Lemma A.1(2,3) we have

1 ≠ ^{e}[σ̄, τ̄] ∈ T_{(e+1)} \ T_{(p−1)}.

Moreover, since N is abelian of index p, we deduce from ^{e+1}[σ̄, τ̄] = 1 that ^{e}[σ̄, τ̄] ∈ Z(T). We obtain that some invariant t_{i} for 3 ≤ i < p is nonzero. Now if Γ were an absolute Galois group of a field F, then setting E = Fix(Δ) we obtain T = T_{E/F}, and then we have a contradiction to Theorem A.2.

Now assume additionally that there exists a closed normal subgroup Λ ⊂ Δ of Γ such that Γ/Λ ≃ ℤ/p^{2}ℤ and e = 1. Let σ̃ denote the image of σ in T. We have a commutative diagram

\xymatrix{1 \ar[r] & N \ar[r] \ar@{->>}[d] & T \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{1} \ar[r] & H \ar[r] & G \ar[r] & 1}

with an image of σ̃ in H of order p^{2}. Hence by Lemma A.1(6), the invariant u of T is equal to 1. Then as before we deduce that some invariant t_{i}, 2 ≤ i < p, is nonzero. Again using Theorem A.2, Γ is not the absolute Galois group of a field.

(2). We translate the conditions into T and N as before, obtaining two elements [σ̄, τ̄_{i}], i = 1, 2, which generate distinct subgroups of Z(T) ∩ T_{(2)} with trivial intersection with T_{(p)}. We deduce that the sum of t_{i}, 2 ≤ i < p, is at least two. If some t_{i}, 3 ≤ i < p, is at least one, then Theorem A.2 tells us that Γ is not the absolute Galois group of a field. Otherwise we have t_{2} ≥ 2, and then again by Theorem A.2, Γ is not the absolute Galois group of a field.

(3). We translate the conditions into T and N as before, obtaining σ̄ ∉ N, τ̄ ∈ N, and

σ̄^{p} ∈ ^{2}[σ̄, N].

By Lemma A.1(2,3) we have σ̄^{p} ⊂ T_{(3)}, and from Lemma A.1(5) we deduce that T^{p} ⊂ T_{(3)}. Hence the invariant u of T is greater than or equal to 3. Again by Theorem A.2, Γ is not the absolute Galois group of a field. □

Proof of Corollary to Theorem A.1. Let Δ be the closed subgroup of Γ generated as a normal subgroup by σ_{1}^{p}, σ_{2}, and σ_{i} for i ∈ I, and let Λ be the closed subgroup of Γ generated as a normal subgroup by σ_{1}^{p^{2}}, σ_{2}, and σ_{i} for i ∈ I. Set also T = Γ/Δ^{(2)}. Examining the quotient of Γ obtained by trivializing σ_{2} and each σ_{i}, i ∈ I, we see that Δ is maximal and Γ/Λ ≃ ℤ/p^{2}ℤ.

Now let e = f − 1. By passing from Γ to T using bars for denoting images of elements of Γ in T, we see that ^{e}[σ̄_{1}, σ̄_{2}] ∉ ^{p−1}[σ̄_{1}, Δ̄] in T. On the other hand, ^{e+1}[σ_{1}, σ_{2}] = ^{f}[σ_{1}, σ_{2}] = 1 in T. By Theorem A.1(1), Γ is not an absolute Galois group. □

3. Analogues of Theorems of Artin-Schreier and Becker

Recall that by Artin-Schreier there are no elements of order p in the absolute Galois groups of a field F, unless p = 2 and F is formally real. Becker proved that the same holds for maximal pro-p-quotients of absolute Galois groups [Be]. In this section we show that in certain small quotients of absolute Galois groups, there are no non-central elements of order p unless p = 2 and the base field F is formally real.

Let p be a prime and F a field containing a primitive pth root of unity. We define the following fields associated to F (see [AGKM] and [MSp1]):

• F^{(2)}: the compositum of all cyclic extensions of F of degree p • F^{(3)}: the compositum of all cyclic extensions of F^{(2)} of degree

p which are Galois over F

• F^{(2)}^{(2)}: the compositum of all cyclic extensions of F^{(2)} of degree p.

Then we set W_{F} = Gal(F^{(3)}/F) and V_{F} = Gal(F^{(2)}^{(2)}/F). Observe that for a cyclic extension E/F of degree p, T_{E/F} = Gal(E^{(2)}/F).

Theorem A.4. Let p be a prime and F a field containing a primitive pth root of unity ξ_{p}. The following are equivalent:

(1) There exists σ ∈ V_{F} \ Φ(V_{F}) of order p

(2) There exists σ ∈ W_{F} \ Φ(W_{F}) of order p

(3) There exists σ ∈ V_{F} \ Φ(V_{F}) such that for each cyclic E/F of degree p its image σ_{E/F} ∈ T_{E/F} has order at most p.

(4) p = 2 and F is formally real.

If these conditions hold, the elements whose square roots are fixed by σ form an ordering of F.

Remark. The fact that W_{F} in particular is sufficiently large to encode field-theoretic properties was brought to light in [MSp2], which explained the Witt ring of a field F in terms of W_{F} when p = 2.

If W_{F} is a finite group, we may say more. Recall that for a group W an element in a cohomology ring H^{∗}(W, 𝔽_{p}) is called essential if its restriction to all proper subgroups is zero. Recall also that H^{∗}(W, 𝔽_{p}) is Cohen-Macaulay if it is free and finitely generated over a polynomial subalgebra of H^{∗}(W, 𝔽_{p}).

Corollary. Suppose that either p is an odd prime or p = 2 and F is not formally real. Suppose also that F^{×}/F^{×p} is finite. Then the cohomology ring of W_{F} with 𝔽_{p}-coefficients is Cohen-Macaulay and contains no nonzero essential elements.

Proof. By [AK, Theorem 2.1], the result follows if we show that our hypotheses imply that W_{F} is finite and that every element σ of order p in W_{F} is central.

Since F^{×}/F^{×p} is finite, F^{(2)} is a finite Galois extension of F. Considering a chain F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n} = F^{(2)}, where [F_{i+1} : F_{i}] = p for i = 0, …, n − 1, and using [LMS, Theorems 1 and 2], we see that F^{(2)×}/(F^{(2)×})^{p} is also finite. Thus F^{(3)} is a finite Galois extension and W_{F} is finite. From the construction of F^{(3)} we obtain that the Frattini subgroup Φ(W_{F}) of W_{F} is Gal(F^{(3)}/F^{(2)}). Recall that F^{(3)} is generated by cyclic extensions N/F^{(2)} of degree p such that N/F is Galois. Since Gal(F^{(2)}/F) is a finite p-group acting on the cyclic group Gal(N/F^{(2)}) of order p, we deduce that Gal(N/F^{(2)}) lies in the center of Gal(N/F). Therefore Φ(W_{F}) is a subgroup of the center Z(W_{F}) of W_{F}.

Now let σ be any element of order p as in Theorem A.2. Then σ ∈ Φ(W_{F}) ⊂ Z(W_{F}), as required. □

Remark. The case p = 2 was obtained in [AKM, §3].

Proof of Theorem A.4. (1) ⟹ (2) and (1) ⟹ (3). Observe first that F^{(3)} ⊂ F^{(2)}^{(2)}. Hence we have a natural surjection V_{F} ↠ W_{F}. Assume that σ ∈ V_{F} \Φ(V_{F}) has order p. Then its image in W_{F} \Φ(W_{F}) also has order p. Moreover, for any cyclic extension E/F of degree p, the image σ_{E/F} of σ in T_{E/F} has order at most p, as follows. Since E^{(2)} is contained in F^{(2)}^{(2)} we see that σ_{E/F}^{p} = 1 in T_{E/F}.

(2) ⟹ (4). Suppose that σ ∈ W_{F} \ Φ(W_{F}) has order p. As in the proof of Theorem A.2, let a ∈ F^{×} \ F^{×p} be arbitrary such that F (\sqrt[p]{a}) ⊄ F (ξ_{p^{2}}). Set K_{a} := F (\sqrt[p]{a}, ξ_{p^{2}}) ⊂ F^{(2)}. Then [K_{a} : F (ξ_{p^{2}})] = p and L_{a} := F ( \sqrt[p^{2}]{a}, ξ_{p^{2}}) is a cyclic extension of degree p^{2} of F (ξ_{p^{2}}). Moreover, L_{a}/F is a Galois extension of F and L_{a} ⊂ F^{(3)} since Gal(L_{a}/F) is a central extension of degree ≤ p of Gal(K_{a}/F). Now if σ(\sqrt[p]{a}) ≠ \sqrt[p]{a} then σ has order p^{2} in W_{F}. Hence σ fixes all \sqrt[p]{a} for a ∈ F^{×} with F ≠ F (\sqrt[p]{a}) ⊄ F (ξ_{p^{2}}). Since σ ∉ Φ(W_{F}), there must exist a cyclic extension E ⊂ F^{(2)} of F which is not fixed by σ. We deduce that ξ_{p^{2}} ∉ F and so E = F (ξ_{p^{2}}). If p is odd then L := F (ξ_{p^{3}}) is a cyclic extension of F of degree p^{2} and σ restricts to a generator of Gal(L/F) ≃ ℤ/p^{2}ℤ, again a contradiction, whence p = 2. Now let s ∈ W_{F} \ Φ(W_{F}) be an element of order 2. By [MSp1, proof of Theorem 2.7], the elements of F whose square roots are fixed by s form an ordering of F. Hence (2) ⟹ (4).

(4) ⟹ (1). Suppose that (4) holds. By [Be, Satz 3], there exists an ordering of F whose square roots are fixed by some element s of order 2 in the maximal pro-2 quotient G_{F} (2) of G_{F}. Then the restriction of s to F^{(2)}^{(2)} is the required element σ ∈ V_{F} \ Φ(V_{F}) of order 2. Hence (4) ⟹ (1).

(3) ⟹ (2). Let σ ∈ V_{F} \ Φ(V_{F}) such that for each cyclic extension E/F of degree p the image σ_{E/F} of σ in T_{E/F} has order at most p. Let E = F (\sqrt[p]{a}) such that σ acts nontrivially on \sqrt[p]{a} and L_{a} = F ( \sqrt[p^{2}]{a}, ξ_{p^{2}}). As above, since the restriction of σ_{E/F} to L_{a} ⊂ E^{(2)} is not of order p^{2}, we deduce that p = 2 and \sqrt{−1} ∉ F. Hence F is not quadratically closed. If F is real closed then there exists precisely one extension E/F of degree 2, namely F^{(2)}, and F^{(2)} = F^{(3)} = F^{(2)}^{(2)}, whence W_{F} = T_{E/F} = V_{F}. Otherwise, F^{(3)} is a compositum of extensions L/F such that Gal(L/F) is either a cyclic group of order 4 or a dihedral group of order 8. (See [MSp2, Corollary 2.18].) On the other hand, each such L lies in E^{(2)} for a suitable quadratic extension E/F: each L may be obtained as a Galois closure of E(\sqrt{γ}) for some [E : F] = 2 and γ ∈ E^{×}. Therefore the restrictions σ_{L/F} of σ to the extensions L/F have order ≤ 2, and so the restriction of σ_{F^{(3)}/F} of σ to F^{(3)}/F has order 2. Hence (3) ⟹ (2). □

4. Families of Pro-p-Groups That are Not Absolute Galois Groups

Theorem A.5. Let p > 3 be prime. There exists a pro-p-ℤ_{p} operator group Ω such that no group of the form

Γ := ((Ω ⋆ Σ) ⋊ ℤ_{p})/E,

where

• Σ is any pro-p-group with trivial ℤ_{p}-action, and

• E is any normal closed subgroup of (Ω ⋆ Σ) ⋊ ℤ_{p} such that

E ⊂ ((Ω ⋆ Σ) ⋊ pℤ_{p})^{(3)},

is an absolute Galois group.

Here ⋆ denotes the free product in the category of pro-p-groups. (Recall that R^{(n)} denotes the nth term of the p-descending series of a pro-p-group R; see section 1.)

The proof of Theorem A.5 relies on [LMS, Theorem 1] and the existence of pro-p-ℤ_{p} operator groups Ω with certain cohomological properties, established in the following proposition. We prove this proposition in sections 5 and 6. Recall that for a pro-p-group Γ, the decomposable subgroup of H^{2}(Γ, 𝔽_{p}) is defined to be the subgroup generated by cup products of elements of H^{1}(Γ, 𝔽_{p}):

H^{2}(Γ, 𝔽_{p})^{dec} = H^{1}(Γ, 𝔽_{p}).H^{1}(Γ, 𝔽_{p}).

We say that H^{2}(Γ, 𝔽_{p}) is decomposable if H^{2}(Γ, 𝔽_{p}) = H^{2}(Γ, 𝔽_{p})^{dec}.

Proposition A.3. Let p > 3 be prime and C be a cyclic group of order p.

(1) There exists a torsion-free pro-p-C operator group Ω_{1} such that as 𝔽_{p}C-modules, H^{1}(Ω_{1}, 𝔽_{p}) ≃ M_{p} and

H^{2}(Ω_{1}, 𝔽_{p}) = H^{2}(Ω_{1}, 𝔽_{p})^{dec} ≃ M_{p−1} ⊕ ⨁_{i=1}^{(p−3)/2} M_{p}.

(2) There exists a torsion-free pro-p-C operator group Ω_{2} such that as 𝔽_{p}C-modules, H^{1}(Ω_{2}, 𝔽_{p}) ≃ M_{2} ⊕ M_{p} and

H^{2}(Ω_{2}, 𝔽_{p}) = H^{2}(Ω_{2}, 𝔽_{p})^{dec} ≃ M_{1} ⊕ M_{3} ⊕ ⨁_{i=1}^{(p+1)/2} M_{p}.

Here we use M_{i} to denote the unique cyclic 𝔽_{p}C-module of dimension i, and here and in the remaining sections we write the action of 𝔽_{p}C multiplicatively instead of exponentially.

In what follows we freely use basic properties of the transgression map found in [NSW, Chapters 2 and 3], and we abbreviate this map by tra. We denote by inf and res the inflation and restriction maps of group cohomology, respectively. Given a free pro-p-group V and a prop-group Δ, we say that a surjection V ↠ Δ is a minimal presentation of Δ if inf : H^{1}(Δ, 𝔽_{p}) → H^{1}(V, 𝔽_{p}) is an isomorphism.

Theorem A.6. Let Δ be a pro-p-group, V a free pro-p-group, and 1 → R → V → Δ → 1 a minimal presentation of Δ. Then we have natural maps

(1) H^{2}(Δ, 𝔽_{p}) ≃ (R/(R^{p}[R, V]))^{∨}

(2) H^{2}(Δ, 𝔽_{p})^{dec} ≃ (R/(R ∩ V^{(3)}))^{∨}.

Proof. Set Δ^{[2]} := Δ/Δ^{(2)} and V^{[3]} := V/V^{(3)}. Since the presentation is minimal, Δ^{[2]} ≃ V/V^{(2)}. We then have the following commutative diagram

[figure: 1; R; V; Δ; 1; 1; \frac{V^{(2)}}{V^{(3)}}; V^{[3]}; Δ^{[2]}; 1]

Here the middle and right-hand vertical maps are the natural projections. Since the presentation is minimal, R ⊂ V^{(2)} and the left-hand vertical map is the natural factor map.

Passing to the natural maps induced by the Hochschild-Serre-Lyndon spectral sequence with coefficients in 𝔽_{p}, we obtain the commutative diagram

\xymatrix{H^{1}(Δ^{[2]}) \ar[r]^{inf} \ar[d] & H^{1}(V^{[3]}) \ar[r]^{res} \ar[d] & H^{1} (\frac{V^{(2)}}{V^{(3)}} )^{Δ^{[2]}} \ar[r]^{tra} \ar[d] & H^{2}(Δ^{[2]}) \ar[r] \ar[d] & … \\ H^{1}(Δ) \ar[r]^{inf} & H^{1}(V) \ar[r]^{res} & H^{1}(R)^{Δ} \ar[r]^{tra} & H^{2}(Δ) \ar[r] & 0.}

(Recall that H^{2}(V, 𝔽_{p}) = 0 since V is a free pro-p-group.) Since the extension

1 → V^{(2)}/V^{(3)} → V^{[3]} → Δ^{(2)} → 1

is central, we deduce that H^{1}(V^{(2)}/V^{(3)}, 𝔽_{p})^{Δ^{[2]}} = H^{1}(V^{(2)}/V^{(3)}, 𝔽_{p}). Additionally using the fact that the inflation map on the first cohomology group in each row of the previous diagram is an isomorphism, we may extract the following commutative square, with the rightmost transgression map an isomorphism:

\xymatrix{H^{1} (\frac{V^{(2)}}{V^{(3)}} , 𝔽_{p}) \ar[d]^{tra} \ar[r] & H^{1}(R, 𝔽_{p})^{Δ} \ar[d]^{tra}_{≃} \\ H^{2}(Δ^{[2]}, 𝔽_{p}) \ar[r] & H^{2}(Δ, 𝔽_{p})}

Since H^{1}(R, 𝔽_{p})^{Δ} ≃ (R/R^{p}[R, V])^{∨}, we have (1).

The leftmost transgression map, however, is also an isomorphism by [Ho, 1.1 and proof]. Now consider the natural map R/R^{p}[R, V] → V^{(2)}/V^{(3)} of abelian topological groups of exponent p. The image of the natural map

H^{1}(V^{(2)}/V^{(3)}, 𝔽_{p}) → H^{1}(R, 𝔽_{p})^{Δ}

is H^{1}(RV^{(3)}/V^{(3)}, 𝔽_{p}). We then factor the horizontal maps of the commutative square into homomorphisms followed by injections:

\xymatrix{H^{1} (\frac{V^{(2)}}{V^{(3)}} , 𝔽_{p}) \ar[d]^{tra}_{≃} \ar[r] & H^{1}(\frac{RV^{(3)}}{V^{(3)}} , 𝔽_{p}) \ar@{^{(}->}[r] & H^{1}(R, 𝔽_{p})^{Δ} \ar[d]^{tra}_{≃} \\ H^{2}(Δ^{[2]}, 𝔽_{p}) \ar[r] & H^{2}(Δ, 𝔽_{p})^{dec} \ar@{^{(}->}[r] & H^{2}(Δ, 𝔽_{p}).}

We obtain isomorphisms

H^{1}(R/(R ∩ V^{(3)}), 𝔽_{p}) ≃ H^{1}(RV^{(3)}/V^{(3)}, 𝔽_{p}) ≃ H^{2}(Δ, 𝔽_{p})^{dec}.

Hence H^{2}(Δ, 𝔽_{p})^{dec} ≃ (R/(R ∩ V^{(3)}))^{∨}, and we have proved (2). □

Remark. The fact that R^{p}[R, V] ≃ R∩V^{(3)} if and only if H^{2}(Δ, 𝔽_{p}) is decomposable was proved earlier in the case p = 2 in [GM, Theorem 2] and [MSp2, §5].

Theorem A.7. Let Γ and H be pro-p-groups with maximal closed subgroups Δ and N, respectively, and α : Γ → H a surjection with α(Δ) = N and ker α ⊂ Δ^{(3)}. Write G for Γ/Δ ≃ H/N. Then as 𝔽_{p}G-modules

H^{2}(Δ, 𝔽_{p})^{dec} ≃ H^{2}(N, 𝔽_{p})^{dec}.

Proof. We shall prove that H^{2}(Δ, 𝔽_{p})^{dec} ≃ H^{2}(N, 𝔽_{p})^{dec} under a natural isomorphism, and from the proof it will follow that the isomorphism is 𝔽_{p}G-equivariant. Let

1 → R → V ⟶^{θ} Δ → 1

be a minimal presentation of Δ with V a free pro-p-group. Since ker α ⊂ Δ^{(3)} and θ(V^{(3)}) = Δ^{(3)}, we may choose a section W ⊂ V^{(3)} of ker α under the surjection θ. We then obtain a presentation of N = α(Δ) as follows:

1 → RW → V ⟶^{ψ} N → 1

where ψ = α ∘ θ. Since RW ⊂ V^{(2)}, this presentation is minimal. By Theorem A.6(2) we have natural isomorphisms

H^{2}(Δ, 𝔽_{p})^{dec} ≃ (RV^{(3)}/V^{(3)})^{∨} ≃ (RW V^{(3)}/V^{(3)})^{∨} ≃ H^{2}(N, 𝔽_{p})^{dec}.
□

Lemma A.6. Let p be prime and Γ a nonfree pro-p-group which is the absolute Galois group of a field F. Then char F ≠ p and a primitive pth root of unity ξ_{p} lies in F.

Proof. Since Γ is not free, then by Witt’s Theorem (see [Koc, Theorem 9.1] and [JLY, pages 207-210]), char F ≠ p. If F does not contain a primitive pth root of unity, then F (ξ_{p}) ⊂ F^{s} and Gal(F (ξ_{p})/F) is a quotient of the absolute Galois group Γ, while ([F (ξ_{p}) : F], p) = 1, a contradiction. Hence ξ_{p} ∈ F. □

Theorem A.8. Let p > 3 be prime. Suppose that Γ is a pro-p-group and Δ is a maximal closed subgroup of Γ.

If the 𝔽_{p}(Γ/Δ)-module H^{2}(Δ, 𝔽_{p})^{dec} contains a cyclic summand of dimension i with 3 ≤ i < p, then Γ is not an absolute Galois group.

Moreover, if Γ contains a normal closed subgroup Λ ⊂ Δ with Γ/Λ ≃ ℤ/p^{2}, then we may take 2 ≤ i < p in the same statement.

Proof. Suppose that Γ and Δ are pro-p-groups satisfying the hypotheses. Contrary to our statement, assume that Γ = G_{F} for some field F. Since Δ is of index p, we obtain that Δ = G_{E} for some Galois extension E/F of degree p. Because H^{2}(Δ, 𝔽_{p}) is nonzero, G_{E} is not a free pro-p-group and therefore neither is G_{F} [S, Corollary 3, §I.4.2]. Lemma A.6 tells us that char F ≠ p and that a primitive pth-root of unity ξ_{p} lies in F. By [MeSu, Theorem 11.5], we obtain that H^{2}(Δ, 𝔽_{p}) is decomposable. Therefore by [LMS, Theorem 1], H^{2}(Δ, 𝔽_{p})^{dec} contains no cyclic 𝔽_{p}(Γ/Δ)-summand of dimension i with 3 ≤ i < p, a contradiction. Moreover, if Λ ⊂ N is a normal closed subgroup and Γ/Λ ≃ ℤ/p^{2}ℤ, then by Albert’s criterion [A], ξ_{p} ∈ N_{E/F} (E). Letting E = F (\sqrt[p]{a}) for some a ∈ F^{×}, we obtain (a).(ξ_{p}) = 0 as an element of H^{2}(Γ, 𝔽_{p}). Then by [LMS, Theorem 1], H^{2}(Δ, 𝔽_{p})^{dec} contains no cyclic 𝔽_{p}(Γ/Δ)-summand of dimension 2, again a contradiction. □

Corollary A.9. Let p > 3 be a prime. Suppose that Γ and H are prop-groups with respective maximal subgroups Δ and N, and α : Γ → H a surjection with α(Δ) = N and ker α ⊂ Δ^{(3)}. Write G for Γ/Δ ≃ H/N. If either H^{2}(Δ, 𝔽_{p})^{dec} or H^{2}(N, 𝔽_{p})^{dec} contains a cyclic 𝔽_{p}G-summand of dimension i with 3 ≤ i < p, then neither Δ nor H is an absolute Galois group.

Moreover, suppose additionally that H contains a normal closed subgroup Λ ⊂ N with H/Λ ≃ ℤ/p^{2}ℤ. Then if either H^{2}(Δ, 𝔽_{p})^{dec} or H^{2}(N, 𝔽_{p})^{dec} contains a cyclic 𝔽_{p}G-summand of dimension 2, then H is not an absolute Galois group.

Proof. By Theorem A.7 we have that H^{2}(Δ, 𝔽_{p})^{dec} ≃ H^{2}(N, 𝔽_{p})^{dec} as 𝔽_{p}G-modules. The remainder follows from Theorem A.8. □

Proof of Theorem A.5. Let Ω be either of the groups Ω_{i} described in Proposition A.3 and let Σ and Γ be defined as in the statement of the theorem.

Observe that Δ := ((Ω⋆Σ)×pℤ_{p})/E is a maximal closed subgroup of Γ of index p. Let G = Γ/Δ and note that the actions of G and C on Δ are identical. By Corollary A.9 it is enough to show that H^{2}(Δ, 𝔽_{p})^{dec} for E = 1 contains a cyclic 𝔽_{p}G-summand M_{i} with 3 ≤ i < p. Assume then that E = 1.

By [NSW, Theorem 4.1.4] we calculate 𝔽_{p}G-module isomorphisms

H^{1}(Δ, 𝔽_{p}) ≃ H^{1}(Ω, 𝔽_{p}) ⊕ H^{1}(Σ, 𝔽_{p}) ⊕ H^{1}(pℤ_{p}, 𝔽_{p})
H^{2}(Δ, 𝔽_{p}) ≃ H^{2}(Ω, 𝔽_{p}) ⊕ H^{2}(Σ, 𝔽_{p}) ⊕ H^{1}(Ω, 𝔽_{p}) ⊕ H^{1}(Σ, 𝔽_{p}).

(Here we use the fact that H^{2}(pℤ_{p}, 𝔽_{p}) = {0} and that H^{2}(A×B, 𝔽_{p}) = H^{2}(A, 𝔽_{p}) ⊕ H^{2}(B, 𝔽_{p}) ⊕ H^{1}(A, 𝔽_{p}) ⊗ H^{1}(B, 𝔽_{p}).)

By Proposition A.3, H^{2}(Ω, 𝔽_{p}) is decomposable. We deduce that H^{2}(Ω, 𝔽_{p}) is a direct summand of H^{2}(Δ, 𝔽_{p})^{dec}. From Proposition A.3, we obtain that H^{2}(Ω, 𝔽_{p}) contains an 𝔽_{p}G-summand M_{i} with 3 ≤ i ≤ p − 1 and we are done. □

The foregoing proof excludes the families of groups Γ from the class of absolute Galois groups by using the n = 2 case of [LMS, Theorem 1]. When Ω = Ω_{1}, the n = 1 case cannot be used directly to exclude the groups Γ. Moreover, neither can Theorem A.2 be used to exclude the groups Γ, as follows. Setting T = Γ/Δ^{(2)} and using duality (see [RZ, Lemma 2.9.4 and Theorem 2.9.6]), we obtain from Lemma A.1(4) the invariants t_{p} = 1, t_{i} = 0 for 2 ≤ i < p, and t_{1} = 1 + dim_{𝔽_{p}} H^{1}(Σ, 𝔽_{p}). Observing that Λ := ((Ω ⋆ Σ) × p^{2}ℤ_{p})/E is a normal subgroup of index p^{2} with quotient ℤ/p^{2}ℤ and contained in Δ, we deduce that u = 1. But this set of invariants is permitted by Theorem A.2.

When Ω = Ω_{2}, the situation is different. We see that H^{1}(Δ, 𝔽_{p}) does not contain any indecomposable 𝔽_{p}C-modules of dimension i for 3 ≤ i < p and so the groups Γ cannot be excluded by a simple observation of impermissible indecomposable 𝔽_{p}C-summands in the n = 1 case of [LMS, Theorem 1]. However, since u = 1 as in the case of Ω = Ω_{1}, we may deduce that the invariant Υ_{2} of the n = 1 case of [LMS, Theorem 1] is zero, as follows. Suppose Γ is the absolute Galois group of a field F. Since Γ is a pro-p-group which is not free, by Lemma A.6 we deduce that F is of characteristic not p, containing a primitive pth root of unity ξ_{p}. Hence E := Fix(Δ) = F (\sqrt[p]{a}) for some a ∈ F^{×}. But since E/F embeds in a cyclic extension of degree 4 given by Fix(Λ), we have that ξ_{p} ∈ N_{E/F} (E^{×}) by [A] and so (a).(ξ_{p}) = 0, whence Υ_{2} = 0. In a similar manner, T = Γ/Δ^{(2)} violates the conditions of Theorem A.2 since t_{2} ≥ 1 when u = 1.

Remark. We observe that the fact that Ω_{i} ⋊ ℤ_{p}, i = 1, 2, is not an absolute Galois group could also be deduced from the main results in [Koe], using different methods. However, the fact that each ((Ω_{i} ×Σ)⋊ ℤ_{p})/E is not an absolute Galois group does not follow from [Koe].

5. The Group Ω_{1} and its Cohomology

In this section we prove Proposition A.3(1).

Let D = ⟨g | g^{p} = 1⟩ be a cyclic group of order p, and let ℤ_{p}D be the p-adic group ring. Because we will form group extensions of the additive group of ℤ_{p}G, to avoid confusion we write the group multiplicatively and denote it by G, where the element ḡ_{i} of G corresponds to the element g^{i} of ℤ_{p}D. (We reserve the symbol g_{i} for elements of the group extension Ω_{1}, which we define later.) Thus ḡ_{0}, ḡ_{1}, …, ḡ_{p−1} form a topological generating set of G. We shall always read the suffixes mod p, so that for example if i = p − 1 then ḡ_{i+1} denotes ḡ_{0}.

Now let C = ⟨σ | σ^{p} = 1⟩ be another group of order p, acting on G via σ(ḡ_{i}) = ḡ_{i−1}. In this way G and H^{1}(G, ℤ_{p}) are free ℤ_{p}C-modules, and H^{1}(G, ℤ_{p}) has a topological generating set y_{0}, y_{1}, …, y_{p−1} dual to ḡ_{0}, ḡ_{1}, …, ḡ_{p−1}. Observe that σ(y_{i}) = y_{i+1}.

Next let H = ℤ_{p} be a trivial ℤ_{p}C-module with generator h, and let z ∈ H^{1}(H, ℤ_{p}) be dual to h. We define a nonsplit central extension Ω_{1} of H by G as follows. The group H^{2}(G, H) = ⋀^{2} H^{1}(G, H) is a free ℤ_{p}Cmodule of rank p(p−1)/2 with free generators y_{0}y_{j} for 1 ≤ j ≤ (p−1)/2. Consider the element

y := (1 + σ + ⋯ + σ^{p−1})y_{0}y_{1} = y_{0}y_{1} + y_{1}y_{2} + ⋯ + y_{p−1}y_{0}.

We let Ω_{1} be the central extension of H by G corresponding to y ∈ H^{2}(G, H):

1 → H → Ω_{1} → G → 1.

The group Ω_{1} is a torsion-free nilpotent pro-p-group of Hirsch length p+1. Examining the standard correspondence of group extensions with H^{2} gives us that for suitable representatives g_{i} ∈ Ω_{1} with respective images ḡ_{i} ∈ G, we have the relations

[g_{i}, g_{j}] = h, j = i + 1
[g_{i}, g_{j}] = 1, j ≠ i + 1, i ≠ j + 1
[g_{i}, h] = 1, all i.

Observe that the action of C on G may be extended to Ω_{1} by σ(g_{i}) = g_{i−1}, 0 ≤ i < p, and σ(h) = h.

The E_{2} page of the spectral sequence

H^{s}(G, H^{t}(H, ℤ_{p})) ⇒ H^{s+t}(Ω_{1}, ℤ_{p})

is generated by the anticommuting elements y_{i} ∈ E_{2}^{1,0} and z ∈ E_{2}^{0,1}, and the differential d_{2} is given by d_{2}^{0,1}(z) = y. (See [NSW, Theorem 2.1.8].) The same holds for the E_{2} page of the spectral sequence with coefficients in 𝔽_{p}

H^{s}(G, H^{t}(H, 𝔽_{p})) ⇒ H^{s+t}(Ω_{1}, 𝔽_{p}),

and we write ȳ_{i} and z̄ for the reduction mod p of the classes y_{i} and z. The picture below indicates the bases of relevant cohomology groups on this E_{2} page.

z̄ z̄.ȳ_{i} z̄.ȳ_{i}.ȳ_{j}
1 ȳ_{i}  ȳ_{i}.ȳ_{j} ȳ_{i}.ȳ_{j}.ȳ_{k}

First, we observe that d_{2}^{0,1} is injective on E^{0,1} = H^{1}(H, 𝔽_{p}), so that by the Five-Term Exact Sequence [NSW, Proposition 2.1.2], H^{1}(Ω_{1}, 𝔽_{p}) ≃ H^{1}(G, 𝔽_{p}) as 𝔽_{p}C-modules. We obtain then that H^{1}(Ω_{1}, 𝔽_{p}) ≃ M_{p}.

Now we turn to the calculation of H^{2}(Ω_{1}, 𝔽_{p}). We claim first that d_{2}^{1,1} is injective, as follows. It is enough to show that d_{2}^{1,1}(z̄.x̄) = 0 implies that x̄ = 0 for x̄ ∈ H^{1}(G, 𝔽_{p}). Write x̄ = ∑_{i=0}^{p−1} c_{i}ȳ_{i}, c_{i} ∈ 𝔽_{p}. Since p > 3, the set of elements ȳ_{i}.ȳ_{i+2}.ȳ_{i+3}, 0 ≤ i < p, is 𝔽_{p}-independent and may be expanded to an 𝔽_{p}-basis of E_{2}^{3,0} = H^{3}(G, 𝔽_{p}) consisting of threefold products of distinct ȳ_{i}. Consider the coefficient β_{i} of ȳ_{i}.ȳ_{i+2}.ȳ_{i+3} in an expansion of d_{2}^{1,1}(z̄.x̄). Since p > 3, the only consecutive pair of indices in ȳ_{i}.ȳ_{i+2}.ȳ_{i+3} is {i + 2, i + 3}. Such a consecutive pair appears the product of the term ȳ_{i+2}.ȳ_{i+3} in ȳ with the term c_{i}ȳ_{i} of x̄. All other nonzero products of terms of ȳ with terms of x̄ that contain a consecutive pair then contain more than one consecutive pair. Hence β_{i} depends only on c_{i}. If all β_{i} = 0, then each c_{i} = 0 and therefore d_{2}^{1,1} is injective.

Since d_{2}^{1,1} is injective, we obtain that E_{3}^{1,1} = E_{∞}^{1,1} = 0. Since E_{2}^{0,2} = 0 we have E_{∞}^{0,2} = 0. Next observe that ȳ = (σ − 1)^{p−1}ȳ_{0}.ȳ_{1} and so ȳ ∈ H^{2}(G, 𝔽_{p})^{C}. Since E^{2,0} = H^{2}(G, 𝔽_{p}) is the direct sum of free 𝔽_{p}Cmodules M(0, i) ≃ M_{p} on generators ȳ_{0}.ȳ_{i}, 1 ≤ i ≤ (p − 1)/2, we deduce that

E_{∞}^{2,0} = E_{3}^{2,0} = (M(0, 1)/M(0, 1)^{C}) ⊕ ⨁_{i=2}^{(p−1)/2} M(0, i).

We conclude that H^{2}(Ω_{1}, 𝔽_{p}) ≃ M_{p−1} ⊕ \frac{(p−3)}{2}M_{p}.

Finally, since H^{1}(Ω_{1}, 𝔽_{p}) ≃ H^{1}(G, 𝔽_{p}) and H^{2}(Ω_{1}, 𝔽_{p}) is a quotient of E_{2}^{2,0} = H^{2}(G, 𝔽_{p}) = ⋀^{2} H^{1}(G, 𝔽_{p}), which is decomposable, we deduce that H^{2}(Ω_{1}, 𝔽_{p}) is decomposable as well.

6. The Group Ω_{2} and its Cohomology

In this section we prove Proposition A.3(2).

As before let D = ⟨g | g^{p} = 1⟩ be a cyclic group of order p, G = ℤ_{p}D the p-adic group ring with topological generating set ḡ_{i} := g^{i} for 0 ≤ i < p, C = ⟨σ | σ^{p} = 1⟩ another group of order p acting on G via σ(ḡ_{i}) = ḡ_{i−1}, and H^{1}(G, ℤ_{p}) the resulting free ℤ_{p}C-module with topological generating set y_{0}, y_{1}, …, y_{p−1} dual to ḡ_{0}, ḡ_{1}, …, ḡ_{p−1}. Again σ(y_{i}) = y_{i+1}.

Now let H = ⟨h | h^{p} = 1⟩ be another cyclic group of order p and ℤ_{p}H be its p-adic group ring. As with ℤ_{p}D, we write ℤ_{p}H multiplicatively, denoting h^{i} by h_{i}. We then set H to be the subgroup of ℤ_{p}H generated by the elements h_{i}h_{j}^{−1}. Then H and H^{1}(H, ℤ_{p}) are free ℤ_{p}-modules of rank p − 1. Let z_{0}, z_{1}, …, z_{p−1} ∈ H^{1}(H, ℤ_{p}) be the images under the surjection H^{1}(ℤ_{p}H, ℤ_{p}) ↠ H^{1}(H, ℤ_{p}) of the topological generating set of H^{1}(ℤ_{p}H, ℤ_{p}) dual to h_{0}, h_{1}, …, h_{p−1} ∈ ℤ_{p}H. Since H is generated by elements of the form h_{i}h_{j}^{−1}, the elements z_{i} satisfy the relation z_{0}+z_{1}+⋯+z_{p−1} = 0. Moreover, the set {h_{0}h_{p−1}^{−1}, h_{1}h_{p−1}^{−1}, …, h_{p−2}h_{p−1}^{−1}} is dual to the set {z_{0}, z_{1}, …, z_{p−2}}.

We define a nonsplit extension Ω_{2} of H by G as follows. Writing H as a free ℤ_{p}-module with ℤ_{p}-base h_{i}h_{p−1}^{−1}, 0 ≤ i < p − 1, we have

H^{2}(G, H) = ⨁_{i=0}^{p−2} H^{2}(G, ⟨h_{i}h_{p−1}^{−1}⟩) ≃ ⨁_{i=0}^{p−2} (⋀^{2} H^{1}(G, ℤ_{p})) .

With these identifications, consider the element y defined as

(−y_{0}.y_{1} + 3y_{1}.y_{2} − 3y_{2}.y_{3} + y_{3}.y_{4}, −y_{1}.y_{2} + 3y_{2}.y_{3} − 3y_{3}.y_{4} + y_{4}.y_{5},

…, −y_{p−2}.y_{p−1} + 3y_{p−1}.y_{0} − 3y_{0}.y_{1} + y_{1}.y_{2}) ∈ ⨁_{i=0}^{p−2} H^{2}(G, ⟨h_{i}h_{p−1}^{−1}⟩).

We let Ω_{2} be the central extension of H by G corresponding to y ∈ H^{2}(G, H):

1 → H → Ω_{2} → G → 1.

The group Ω_{2} is a torsion-free nilpotent pro-p-group of Hirsch length 2p − 1. For suitable representatives g_{i} ∈ Ω_{2} with respective images ḡ_{i} ∈ G, we have the relations

[g_{i}, g_{i+1}] = (h_{i−3}h_{p−1}^{−1})(h_{i−2}h_{p−1}^{−1})^{−3}(h_{i−1}h_{p−1}^{−1})^{3}(h_{i}h_{p−1}^{−1})^{−1}
[g_{i}, g_{j}] = 1,  j ≠ i + 1
[g_{i}, h_{j}h_{k}^{−1}] = 1,  all i, j, k
[h_{i}h_{j}^{−1}, h_{k}h_{l}^{−1}] = 1,  all i, j, k, l.

We check that the action of C on ḡ_{i} may be extended to an action of C on the abelian pro-p-group H and then on Ω_{2}, via σ(g_{i}) = g_{i−1}, σ(h_{i}h_{j}^{−1}) = h_{i−1}h_{j−1}^{−1}, 0 ≤ i, j < p. In particular, H^{1}(H, 𝔽_{p}) is an 𝔽_{p}C-module isomorphic to M_{p−1}.

The E_{2} page of the spectral sequence

H^{s}(G, H^{t}(H, ℤ_{p})) ⇒ H^{s+t}(Ω_{2}, ℤ_{p})

is generated by the anticommuting elements y_{i} ∈ E_{2}^{1,0} and z_{j} ∈ E^{0,1}, and the differential d_{2} is given by

d_{2}^{0,1}(z_{i}) = −y_{i}.y_{i+1} + 3y_{i+1}.y_{i+2} − 3y_{i+2}.y_{i+3} + y_{i+3}.y_{i+4} = (σ − 1)^{3}y_{i}y_{i+1}.

(See [NSW, Theorem 2.1.8].) The same holds for the E_{2} page of the spectral sequence with coefficients in 𝔽_{p}

H^{s}(G, H^{t}(H, 𝔽_{p})) ⇒ H^{s+t}(Ω_{2}, 𝔽_{p}),

and we write ȳ_{i} and z̄_{j} for the reduction mod p of the classes y_{i} and z_{j}. The picture below indicates the bases of relevant cohomology groups on this E_{2} page.

z̄_{i}.z̄_{j} z̄_{i}.z̄_{j}.ȳ_{k}
z̄_{i}  z̄_{i}.ȳ_{j}  z̄_{i}.ȳ_{j}.ȳ_{k}
1  ȳ_{i}  ȳ_{i}.ȳ_{j}  ȳ_{i}.ȳ_{j}.ȳ_{k}

First, we consider d_{2}^{0,1} on E^{0,1} = H^{1}(H, 𝔽_{p}) ≃ M_{p−1}. The group H^{2}(G, 𝔽_{p}) ≃ ⋀^{2} H^{1}(G, 𝔽_{p}) is a direct sum of free ℤ_{p}C-modules M(0, i), 1 ≤ i ≤ (p − 1)/2, each with ℤ_{p}C-module generator y_{0}.y_{i}. For each i in 1 ≤ i ≤ (p − 1)/2, the module M(0, i) consists of the span of the twofold products y_{j}.y_{i+j}, 0 ≤ j < p. From im d_{2}^{0,1} ⊂ M(0, 1) we deduce that im d_{2}^{0,1} = (σ−1)^{3}M(0, 1). Since H^{1}(H, 𝔽_{p}) ≃ M_{p−1} we find that d_{2}^{0,1} has kernel (σ − 1)^{p−3}M_{p−1} ≃ M_{2}, image M_{p−3}, and cokernel M_{3} ⊕ \frac{p−3}{2}M_{p}.

We may now calculate H^{1}(Ω_{2}, 𝔽_{p}). From the Five-Term Exact Sequence we have that H^{1}(Ω_{2}, 𝔽_{p}) is an extension of the free 𝔽_{p}C-module H^{1}(G, 𝔽_{p}) by ker d_{2}^{0,1} ≃ M_{2}. Since free 𝔽_{p}C-modules are injective (see [C, Theorem 11.2]), we deduce that H^{1}(Ω_{2}, 𝔽_{p}) ≃ M_{2} ⊕ M_{p}.

Next we compute the kernel of d_{2}^{1,1}. Multiplying elements in ker d_{2}^{0,1} by arbitrary elements of E_{2}^{1,0} gives elements in ker d_{2}^{1,1}. Since ker d_{2}^{0,1} ≃ M_{2} and E_{2}^{1,0} ≃ M_{p}, we have that ker d_{2}^{1,1} contains a 𝔽_{p}C-submodule isomorphic to M_{2} ⊗ M_{p} ≃ M_{p} ⊕ M_{p}. We claim that ker d_{2}^{1,1} is in fact isomorphic to M_{p} ⊕ M_{p}, as follows. The number of cyclic 𝔽_{p}Csummands of an arbitrary 𝔽_{p}C-module M is equal to dim_{𝔽_{p}} M^{C} (see [LMS, Proposition 2]). Moreover, E_{2}^{1,1} = H^{1}(G, 𝔽_{p}) ⊗ H^{1}(H, 𝔽_{p}) ≃ M_{p} ⊗ M_{p−1} ≃ (p − 1)M_{p}, and so (E_{2}^{1,1})^{C} = (σ − 1)^{p−1}E_{2}^{1,1}. Therefore it suffices to prove that the kernel d_{2}^{1,1} on the image of

N_{σ} := 1 + σ + ⋯ + σ^{p−1} = (σ − 1)^{p−1}

on E_{2}^{1,1} has dimension at most 1.

Let

ν = ∑_{j=0}^{p−1} α_{j} ∑_{k=0}^{p−1} z̄_{k}.ȳ_{j+k} = ∑_{i,j} α_{j−i}z̄_{i}.ȳ_{j},

be an arbitrary element of E_{2}^{1,1}, where α_{i} ∈ 𝔽_{p}. Since ∑_{k=0}^{p−1} z̄_{k} = 0, we may add the same quantity to each of the α_{j} and leave ν unchanged. Therefore we assume that α_{0} = 0, and the remaining α_{j} are uniquely defined. We calculate

d_{2}^{1,1}(ν) = ∑_{i,j} α_{j−i}[(σ − 1)^{3}(ȳ_{i}.ȳ_{i+1})].ȳ_{j}

= ∑_{i,j} ( − α_{j−i} + 3α_{j−i−1} − 3α_{j−i−2} + α_{j−i−3})ȳ_{i}.ȳ_{i+1}.ȳ_{j}.

If d_{2}^{1,1}(ν) = 0, the coefficient of each ȳ_{i}.ȳ_{j}.ȳ_{k} has to be zero. We must be careful, however, because some monomials occur in more than one way, and monomials with repeated indices are zero.

Recalling that α_{0} = 0, suppose that we are further given α_{1} and α_{2} for an element ν in the kernel. Then an examination of the coefficient of ȳ_{0}.ȳ_{1}.ȳ_{3} shows that −α_{0} + 3α_{1} − 3α_{2} + α_{3} = 0, and so α_{3} is determined. Proceeding inductively on i, the coefficient of ȳ_{0}.ȳ_{1}.ȳ_{i} determines α_{i}, except for the last coefficient α_{−1}, because ȳ_{−1}.ȳ_{0}.ȳ_{1} appears in more than one way. Nevertheless, considering the term ȳ_{3}.ȳ_{4}.ȳ_{1}, we find that

−α_{−2} + 3α_{−1} − 3α_{0} + α_{1}.

Since α_{0} = 0 we see that α_{−2} and α_{1} determine α_{−1}. This completes the proof that the kernel of d_{2} on E_{2}^{1,1} is a direct sum of 2 copies of M_{p}. Since free 𝔽_{p}C-modules are injective, ker d_{2}^{1,1} is a direct summand of E_{2}^{1,1} and so E_{∞}^{1,1} = E_{3}^{1,1} = (p − 3)M_{p}.

We now turn to the computation of the kernel of d_{2}^{0,2}. We claim that ker d_{2}^{0,2} = ⋀^{2} ker d_{2}^{0,1} ≃ M_{1}. Since z̄_{i}, 0 ≤ i < p−1, form a basis of E_{2}^{0,1}, the monomials z̄_{i}.z̄_{j}, 0 ≤ i < j < p − 1 form a basis for E_{2}^{0,2}. Similarly, the monomials z̄_{k}.ȳ_{i}.ȳ_{j} for 0 ≤ k < p − 1 and 0 ≤ i < j ≤ p − 1 form a basis for E_{2}^{2,1}. We have

d_{2}^{0,2}(z̄_{i}.z̄_{j}) = z̄_{i}.[(σ − 1)^{3}(ȳ_{j}.ȳ_{j+1})] − z̄_{j}.[(σ − 1)^{3}(ȳ_{i}.ȳ_{i+1})], i < j.

Suppose that γ = ∑_{i<j} β_{i,j}z̄_{i}.z̄_{j} lies in the kernel. We shall show that β_{12} determines the remaining β_{ij}. The coefficient λ_{1,3,4} of z̄_{1}.ȳ_{3}.ȳ_{4} in d_{2}(γ) is determined by the images of monomials z̄_{1}.z̄_{j} for j > 1 since monomials z̄_{i}.z̄_{j} with i < j form a basis; in particular, λ_{1,3,4} is determined by d_{2}^{0,2}(β_{12}z̄_{1}.z̄_{2} + β_{13}z̄_{1}.z̄_{3}). Hence β_{13} is determined by β_{12} and λ_{1,3,4}.

For the remaining β_{ij}, we proceed by induction. Order the monomials z̄_{i}.z̄_{j}, i < j, lexicographically, and assume that β_{kl} is already determined for all z̄_{k}.z̄_{l} < z̄_{i}.z̄_{j} for some fixed 0 ≤ i < j < p − 1. Now consider the coefficient λ_{i,j,j+1} of z̄_{i}.ȳ_{j}.ȳ_{j+1} in d_{2}^{0,2}(γ). The summands β_{kl}z̄_{k}.z̄_{l} of γ contributing to λ_{i,j,j+1} include β_{ij}z̄_{i}.z̄_{j} and various β_{kl}z̄_{k}.z̄_{l} with z̄_{k}.z̄_{l} < z̄_{i}.z̄_{j}. By induction, then, β_{ij} is determined by λ_{i,j,j+1} and β_{kl} for z̄_{k}.z̄_{l} < z̄_{i}.z_{j}. We conclude that ker d_{2}^{0,2} is 1-dimensional and so is ⋀^{2} ker d_{2}^{0,1} ≃ M_{1}.

Next, we observe that E_{3}^{0,2} consists of products of elements of E_{3}^{0,1}, and hence d_{3}^{0,2} : E_{3}^{0,2} → E_{3}^{3,0} is the zero map. We conclude from these calculations above that E_{∞}^{i,j} = E_{3}^{i,j} for i + j ≤ 2, and this portion of the E_{∞} page is as follows:

M_{1}
M_{2} 2M_{p}
M_{1} M_{p} M_{3} ⊕ \frac{p−3}{2}M_{p}

We are now ready to assert that H^{2}(Ω_{2}, 𝔽_{p}) is the direct sum of the E_{∞}^{i,j} with i + j = 2. To see this, we use an automorphism ϕ of Ω_{2} such that ϕ(g_{i}) = g_{i}^{2} for 0 ≤ i ≤ p − 1 and ϕ(h_{i}h_{p−1}^{−1}) = (h_{i}h_{p−1}^{−1})^{4} for 0 ≤ i < p − 1. That ϕ exists one can see from the fact that the commutator map in a nilpotent group of class 2 is a bilinear map. We see that ϕ commutes with the action of σ and the induced action on the spectral sequence satisfies ϕ^{∗}(ȳ_{i}) = 2^{−1}ȳ_{i} and ϕ^{∗}(z̄_{i}) = 4^{−1}z̄_{i}, where a^{−1} denotes the inverse of a modulo p. We calculate that ϕ acts on E_{∞}^{0,2} by multiplication by 16^{−1}, on E_{∞}^{1,1} by multiplication by 8^{−1}, and on E_{∞}^{2,0} by multiplication by 4^{−1}. Since 16^{−1}, 8^{−1} and 4^{−1} are distinct modulo p, we obtain that H^{2}(Ω_{2}, 𝔽_{p}) decomposes as a direct sum of the eigenspaces of ϕ and so H^{2}(Ω_{2}, 𝔽_{p}) ≃ M_{1} ⊕ M_{3} ⊕ \frac{p+1}{2}M_{p}.

Finally, since E_{∞}^{0,2} = ⋀^{2} E_{0,1}, E_{∞}^{1,1} = E_{0,1}.E_{1,0}, and E_{∞}^{2,0} is a quotient of E_{2}^{2,0} = ⋀^{2} E_{0,1}, we deduce that H^{2}(Ω_{2}, 𝔽_{p}) is decomposable.

Remark. The relations among the g_{i} and h_{i}h_{p−1}^{−1} given in the construction of Ω_{2} are minimal since the differential d_{2}^{0,1} on the E_{2} page of the spectral sequence with coefficients in ℤ_{p}, H^{s}(G, H^{t}(H, ℤ_{p})) ⇒ H^{s+t}(Ω_{2}, ℤ_{p}), is injective, as follows. The differential d_{2}^{0,1} is given by d_{2}^{0,1}(z_{i}) = (σ − 1)^{3}(y_{i}.y_{i+1}). Observe that H^{1}(H, ℤ_{p}) ≃ E_{2}^{0,1} is isomorphic as a ℤ_{p}C-module with ℤ_{p}[ξ_{p}], the ring of integers in ℚ_{p}[ξ_{p}], and that σ acts on ℤ_{p}[ξ_{p}] as multiplication by ξ_{p}. We deduce that multiplication by (σ − 1)^{3} is injective. Moreover, the map

e: E_{2}^{0,1} → E_{2}^{2,0},  e(z_{i}) = y_{i}y_{i+1},  0 ≤ i < p − 1,

is injective. Hence d_{2}^{0,1}, as the composition of injective maps, is injective.

Acknowledgement

We thank the referee of [LMS] for suggesting that we illustrate [LMS, Theorem 1] by constructing examples of pro-p-groups which are not absolute Galois groups, for this appendix is the result. The third author would also like to thank Sunil Chebolu, Ajneet Dhillon and Vahid Shirbisheh for stimulating discussions related to this paper.

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38 DAVE BENSON, NICOLE LEMIRE, JÁN MINÁČ, AND JOHN SWALLOW

Department of Mathematical Sciences, University of Aberdeen, Meston Building, King’s College, Aberdeen AB24 3UE, Scotland UK

E-mail address: bensondj@maths.abdn.ac.uk

Department of Mathematics, Middlesex College, University of Western Ontario, London, Ontario N6A 5B7 CANADA

E-mail address: nlemire@uwo.ca

E-mail address: minac@uwo.ca

Department of Mathematics, Davidson College, Box 7046, Davidson, North Carolina 28035-7046 USA

E-mail address: joswallow@davidson.edu