The Whitehead group of (almost) extra-special p-groups with p odd

Serge Bouc^{1} and Nadia Romero^{2}

Abstract: Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup Cl_{1}(ℤP) of SK_{1}(ℤP), in terms of a genetic basis of P. We also introduce a deflation map Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)), for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of SK_{1}(ℤP), when P is an elementary abelian pgroup. Keywords: Whitehead group, almost extra-special p-groups, genetic basis. MSC2010: 19B28, 20C05, 20D15.

Introduction

Whitehead groups were introduced by J.H.C. Whitehead in [19], as an algebraic continuation of his work on combinatorial homotopy. The computation of the Whitehead group W h(G) of a finite group G is in general very hard, and a compendium on the subject is the book by Bob Oliver ([13]) of 1988. Since then, it seems that not much progress has been made on this subject (see however [12], [11], [18], [17]).

Let p be an odd prime number. In this paper, we describe the Whitehead group of all extra-special and almost extra-special p-groups. The main reason for focusing on these p-groups is that, apart from elementary abelian p-groups, they are exactly the finite p-groups all proper factor groups of which are elementary abelian (see e.g. Lemma 3.1 of [8]). In particular these groups appear naturally in various areas as first crucial step in inductive procedures (see [8], [7], or the proof of Serre’s Theorem in Section 4.7 of [3] for examples).

One of the main tools in our method is Theorem 9.5 of [13], which gives a first description, for a finite p-group P, of the subgroup Cl_{1}(ℤP), an essential part of the torsion of W h(P). In addition to this, we make use of biset functor techniques, in two different ways: first we use genetic bases of finite p-groups. They allow for an explicit combinatorial description of the rational irreducible representations of P. This yields a description of Cl_{1}(ℤP), in terms of a genetic basis of P. Next we show that for any normal subgroup N of P, there is a deflation operation Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)). Even if there seems to be no inflation operation in the other direction, which would provide a section of this map, we show that Def_{P/N}^{P} is always surjective.

1CNRS-LAMFA, UPJV, Amiens, France

2DEMAT, UGTO, Guanajuato, Mexico - Partially supported by SEP-PRODEP, project PTC-486

In the case of an extra-special or almost extra-special p-group P, the group Cl_{1}(ℤP) is equal to the torsion part SK_{1}(ℤP) of W h(P), so the computation of the Whitehead group of P comes down to the knowledge of Cl_{1}(ℤ(P/Φ(P))), where Φ(P) is the Frattini subgroup of P, and a detailed analysis of what happens with the unique faithful rational irreducible representation of P. It should be noted that our methods not only give the algebraic structure of the Whitehead group, but allow also for the determination of explicit generators of its torsion subgroup.

The paper is organized as follows: Section 1 is a review of definitions and basic results on Whitehead groups, genetic bases of p-groups, extra-special and almost extra-special p-groups. In particular, the subgroups Cl_{1}(ℤG) and SK_{1}(ℤG) of the Whitehead group W h(G) of a group G are introduced. In Section 2, we give a procedure (Theorem 2.4) to compute Cl_{1}(ℤP) for a finite p-group P (with p odd) in terms of a genetic basis of P. This procedure may be of independent interest, in particular from an algorithmic point of view. Finally, in Section 3, we begin by giving a new proof of the structure of W h(P) for an elementary abelian p-group P, for p odd, and then we come to our main theorems (Theorem 3.17 and Theorem 3.20), which give the structure of W h(P), when P is an extra-special or almost extra-special p-group, for p odd.

1. Preliminaries

Let G be a group. We will write Z(G) for the center of G and G′ for its commutator subgroup. The Frattini subgroup of G is denoted by Φ(G). If H is a subgroup of G, the normalizer of H in G will be denoted by N_{G}(H), and its centralizer by C_{G}(H). If H = ⟨h⟩ for an element h ∈ G, we may also write C_{G}(H) = C_{G}(h).

1.1. About the Whitehead group.

Let R be an associative ring with unit. The infinite general linear group of R, denoted by GL(R), is defined as the direct limit of the inclusions GL_{n}(R) → GL_{n+1}(R) as the upper left block matrix. If, for each n > 0, we denote by E_{n}(R) the subgroup of GL_{n}(R) generated by all elementary n×n-matrices – i.e. all those which are the identity except for one non-zero off-diagonal entry – and we take the direct limit as before for the groups E_{n}(R), we obtain a subgroup of GL(R), denoted by E(R). Whitehead’s Lemma states that E(R) is equal to the derived subgroup of GL(R). The group K_{1}(R) is defined as the abelianization of GL(R), which is then equal to GL(R)/E(R).

If G is a group and we take R as the group ring ℤG, then elements of the form ±g for g ∈ G can be regarded as invertible 1 × 1-matrices over ℤG and hence they represent elements in K_{1}(ℤG). Let H be the subgroup of K_{1}(ℤG) generated by classes of elements of the form ±g with g ∈ G. The Whitehead group of G is defined as W h(G) = K_{1}(ℤG)/H.

If G is finite, the groups K_{1}(ℤG) and W h(G) are finitely generated abelian groups. For the rest of this section, G denotes a finite group.

1.2. Definition. Let D be an integral domain and K be its field of fractions, then SK_{1}(DG) denotes the kernel of the morphism

K_{1}(DG) → K_{1}(KG).

By Theorem 7.4 in [13], SK_{1}(ℤG) is isomorphic to the torsion subgroup of W h(G). Hence, W h(G) is completely determined by SK_{1}(ℤG) and the rank of its free part (i.e. its free rank). According to Theorem 2.6 in [13], this free rank is equal to r − q, where r is the number of non-isomorphic irreducible ℝ-representations of G and q is the number of non-isomorphic irreducible ℚ-representations of G.

1.3. Definition. Consider the ring of p-adic integers ℤ̂_{p}. The group Cl_{1}(ℤG) is defined as the kernel of the localization morphism

l : SK_{1}(ℤG) → ⨁_{p} SK_{1}(ℤ̂_{p}G).

By Theorem 3.9 in [13], SK_{1}(ℤ̂_{p}G) is trivial whenever p does not divide |G|, and l is onto. In particular, SK_{1}(ℤG) sits in an extension

0 ⟶ Cl_{1}(ℤG) ⟶ SK_{1}(ℤG) ⟶ ⨁_{p} SK_{1}(ℤ̂_{p}G) ⟶ 0.

This extension is used by Oliver in [13] to describe SK_{1}(ℤG) in many cases. The important feature of the examples treated in this paper is that they all satisfy that ⨁_{p} SK_{1}(ℤ̂_{p}G) is trivial, and so describing SK_{1}(ℤG) amounts to describing Cl_{1}(ℤG).

1.4. About genetic bases.

A finite p-group Q is called a Roquette p-group if it has normal p-rank 1, i.e. if all its abelian normal subgroups are cyclic. The Roquette p-groups (see [15] or Theorem 4.10 of [10] for details) of order p^{n} are the cyclic groups C_{p^{n}}, if p is odd, and the cyclic groups C_{2^{n}}, the generalized quaternion groups Q_{2^{n}}, for n ⩾ 3, the dihedral groups D_{2^{n}}, for n ⩾ 4, and the semidihedral groups SD_{2^{n}}, for n ⩾ 4, if p = 2. A Roquette p-group Q admits a unique rational irreducible representation Φ_{Q} (see e.g. Proposition 9.3.5 in [5]). 1.5. Definition. Let P be a finite p-group. A subgroup S of P is called genetic if the section S ⊴ N_{P} (S) ⩽ P satisfies

1. The group N_{P} (S)/S is a Roquette group.

2. Let Φ = Φ_{N_{P} (S)/S} be the only faithful irreducible ℚ-representation of N_{P} (S)/S and V = Ind_{N_{P} (S)}^{P}Inf_{N_{P} (S)/S}^{N_{P} (S)}ϕ, then the functor Ind_{N_{P} (S)}^{P}Inf_{N_{P} (S)/S}^{N_{P} (S)} induces an isomorphism of ℚ-algebras

End_{ℚP} V ≅ End_{ℚ(N_{P} (S)/S)}Φ.

Note that the right-hand side algebra is actually a skew field by Schur’s lemma. So End_{ℚP} V is also a skew field, hence V is an indecomposable – that is, irreducible – ℚP -module. 1.6. Notation. Let P be a finite p-group and S be a genetic subgroup of P. We write

V (S) = Ind_{N_{P} (S)}^{P}Inf_{N_{P} (S)/S}^{N_{P} (S)}Φ_{N_{P} (S)/S}.

Then V (S) is an irreducible ℚ-representation of P. Conversely, by Roquette’s Theorem (Theorem 9.4.1 in [5]), for each irreducible ℚ-representation V of P, there exists a genetic subgroup S of P such that V ≅ V (S).

The following theorem characterizes the genetic subgroups of a p-group. First some notation: for a subgroup S of a finite p-group P, let Z_{P} (S) ⩾ S be the subgroup of N_{P} (S) defined by Z_{P} (S)/S = Z(N_{P} (S)/S). In particular Z_{P} (S) = N_{P} (S) if N_{P} (S)/S abelian, e.g. if N_{P} (S)/S is a Roquette p-group for p odd. 1.7. Theorem. [Theorem 9.5.6 in [5]] Let P be a finite p-group and S be a subgroup of P such that N_{P} (S)/S is a Roquette group. Then the following conditions are equivalent:

1. The subgroup S is a genetic subgroup of P.

2. If x ∈ P is such that ^{x}S ∩ Z_{P} (S) ⩽ S, then ^{x}S = S.

3. If x ∈ P is such that ^{x}S ∩ Z_{P} (S) ⩽ S and S^{x} ∩ Z_{P} (S) ⩽ S, then ^{x}S = S.

The next result is part of Theorem 9.6.1 in [5].

1.8. Theorem. Let P be a finite p-group and S and T be genetic subgroups of P. The following conditions are equivalent:

1. The ℚP -modules V (S) and V (T) are isomorphic.

2. There exist x, y ∈ P such that ^{x}T ∩ Z_{P} (S) ⩽ S and ^{y}S ∩ Z_{P} (T) ⩽ T.

3. There exists x ∈ P such that ^{x}T ∩ Z_{P} (S) ⩽ S and S^{x} ∩ Z_{P} (T) ⩽ T.

If these conditions hold, then in particular the groups N_{P} (S)/S and N_{P} (T)/T are isomorphic.

The relation between groups appearing in point 2 is denoted by S _{P} T. The theorem shows that this relation is an equivalence relation on the set of genetic subgroups of P, and we have the following definition.

1.9. Definition. [Definition 9.6.11 in [5]] Let P be a finite p-group. A genetic basis of P is a set of representatives of the equivalence classes of _{P} in the set of genetic subgroups of P.

1.10. Lemma. Let P be a finite p-group and S be a genetic subgroup of P.

1. The kernel of V (S) is equal to the intersection of the conjugates of S in P.

2. In particular V (S) is faithful if and only S intersects Z(P) trivially.

Proof. Denote by Φ the unique rational irreducible representation of the Roquette group N_{P} (S)/S. Then

V (S) = Ind_{N_{P} (S)}^{P}Inf_{N_{P} (S)/S}^{N_{P} (S)}Φ ≅ ⨁_{x∈[P/N_{P} (S)]} x ⊗ Φ,

where [P/N_{P} (S)] is a chosen set of representatives of N_{P} (S)-cosets in P. An element g ∈ P acts trivially on V (S) if and only if it permutes trivially the summands of this decomposition, that is if g^{x} ∈ N_{G}(S) for any x ∈ G, and if moreover g^{x} acts trivially on Φ, which means that g^{x} ∈ S, since Φ is faithful. This proves Assertion 1.

Assertion 2 follows, since

( ⋂_{x∈P} ^{x}S) ∩ Z(P) = ⋂_{x∈P} (^{x}S ∩ Z(P)) = ⋂_{x∈P} ^{x}(S ∩ Z(P)) = S ∩ Z(P),

and since the normal subgroup ⋂_{x∈P} ^{x}S of P is trivial if and only if it intersects Z(P)

trivially.

1.11. Remark. If P is abelian, then a subgroup S of P is genetic if and only if P/S is cyclic. Moreover the relation _{P} is the equality relation in this case, so there is a unique genetic basis of P, consisting of all the subgroups S of P such that P/S is cyclic.

1.12. Remark. Let P be a finite p-group, and S be a subgroup of P such that N_{P} (S) is normal in P. If N_{P} (S)/S is a Roquette group, then S is a genetic subgroup of P. Indeed if N_{P} (S) is normal in P, then N_{P} (S) = N_{P} (^{x}S) for any x ∈ P. Hence xS ⊴ N_{P} (S), and the group ^{x}S · S/S is a normal subgroup of N_{P} (S)/S. It is trivial if and only if it intersects trivially the center Z_{P} (S)/S of N_{P} (S)/S. Then

xS = S ⟺ ^{x}S · S/S = 1 ⟺ ^{x}S · S ∩ Z_{P} (S) = (^{x}S ∩ Z_{P} (S))S = S
⟺ ^{x}S ∩ Z_{P} (S) ⩽ S.

1.13. About extra-special and almost extra-special p-groups.

1.14. Definition. Let p be a prime and P be a finite p-group.

1. The group P is called extra-special if Z(P) = P′ = Φ(P) has order p.

2. The group P is called almost extra-special if P′ = Φ(P) has order p and Z(P) is cyclic of order p^{2}.

Extra-special and almost extra-special p-groups can be classified in the following way.

1.15. Notation. Let H, K and M be groups such that M ⩽ Z(H) and such that there exists an injective map θ : M → Z(K). The central product of H and K with respect to θ will be denoted by H ∗_{θ} K, and simply by H ∗ K if θ is clear from the context.

For any integer r ⩾ 1, we will write H^{∗r} for the central product of r copies of the group H, where M = Z(H), with the convention H^{∗1} = H.

For p ≠ 2, set

M(p) = ⟨x, y | x^{p} = y^{p} = 1, [x^{−1}, y] = [y, x] = ^{y}[y, x]⟩

and

N(p) = ⟨x, y | x^{p^{2}} = y^{p} = 1,^{y}x = x^{1+p}⟩.

1.16. Theorem. Let p be a prime and P be a finite p-group.

1. If P is extra-special, then there exists an integer r ⩾ 1 such that P has order p^{2r+1} and P is isomorphic to only one of the following groups: D_{8}^{∗r} or Q_{8} ∗ D_{8}^{∗(r−1)} if p = 2, and M(p)^{∗r} or N(p) ∗ M(p)^{∗(r−1)} if p ≠ 2.

2. If P is almost extra-special, then there exists an integer r ⩾ 1 such that P has order p^{2r+2} and P is isomorphic to only one of the following groups: D_{8}^{∗r} ∗ C_{4} if p = 2, and M(p)^{∗r} ∗ C_{p^{2}} if p ≠ 2.

Proof. The proof of 1 can be found in Section 5.5 of [10]. As for point 2, one can refer to Sections 2 and 4 of [8].

Observe that if p is odd, the exponent of the group characterizes the isomorphism type of extra-special p-groups, one of them has exponent p and the other one has exponent p^{2}.

If P is an (almost) extra-special group, the quotient P/P′ is elementary abelian, so it can be regarded as a (finite-dimensional) vector space E over the finite field 𝔽_{p}. Moreover, if we take z a generator of P′, then E is endowed with a bilinear form

b : E × E → 𝔽_{p},

that sends an element (u, v) to b(u, v), the element of 𝔽_{p} satisfying [ũ, ṽ] = z^{b(u, v)} for all ũ ∈ u, ṽ ∈ v and u, v ∈ E. This bilinear form is alternating, i.e. b(v, v) = 0 for all v ∈ E, hence it is antisymmetric, i.e. b(u, v) = −b(v, u) for all u, v ∈ E. Section 20 in [9] is concerned with this bilinear form for extra-special groups, but the property of being alternating is called symplectic.

Recall that if f : V ×V → K is a bilinear form on a finite dimensional vector space V over a field K, its left radical V^{⊥} is defined by V^{⊥} = {v ∈ V | f(v, w) = 0 ∀w ∈ V }, and its right radical by ^{⊥}V = {v ∈ V | f(w, v) = 0 ∀w ∈ V }. Clearly V^{⊥} = ^{⊥}V when f is antisymmetric. The rank of f is the codimension of V^{⊥}. The form f is called non-degenerate if V^{⊥} = {0}.

With the help of Lemma 20.4 in [9], we have the following observation.

1.17. Observation. The bilinear form b is non-degenerate if and only if P is extraspecial. If P is almost extra-special, then E^{⊥} = π(Z(P)) is a line in E, where π : P → P/P′ is the projection morphism.

In section 3.2 we will use the following result, which is part of Lemma 2.6 in [7]. For the proof we refer the reader to this reference.

1.18. Lemma. Let P be an (almost) extra-special p-group, and let Q be a non-trivial subgroup of P. Then

1. Q ⊴ P ⇔ P′ ⩽ Q.

2. If Q is not normal in P, then N_{P} (Q) = C_{P} (Q). In particular, it follows that in this case Q is elementary abelian of rank at most r, for the integer r defined as in Theorem 1.16, and we have |Q||C_{P} (Q)| = |P |. Moreover, C_{P} (Q) = Q × U, where U is (almost) extra-special of order |P |/|Q|^{2} or U = Z(P).

2. Cl_{1} of finite p-group algebras for odd p

The goal of this section is to re-write Theorem 9.5 in [13] in terms of genetic bases, in the most possible succinct way. We take the statement of this theorem appearing in Section 1 of [14], which says the following: let p be an odd prime, let P be a finite p-group, and write ℚP ≅ ∏_{i=1}^{k} A_{i}, where A_{i} is simple with irreducible module V_{i} and center K_{i} = End_{ℚP} V_{i}. By Roquette’s Theorem, for each 1 ⩽ i ⩽ k, the field K_{i} is isomorphic to ℚ(ζ_{r_{i}}), where ζ_{r_{i}} is a primitive p^{r_{i}}-th root of unity for some non-negative r_{i}, and A_{i} is isomorphic to a matrix algebra over ℚ(ζ_{r_{i}}).

Consider the abelian group T = ∏_{i=1}^{k}⟨ζ_{r_{i}}⟩. For each h ∈ P, define

ψ_{h} : C_{P} (h) → T, ψ_{h}(g) = (det_{K_{i}}(g, V_{i}^{h}))_{i}

where V_{i}^{h} = {x ∈ V_{i} | hx = x}. Here V_{i}^{h} is viewed as a K_{i}C_{P} (h)-module, so det_{K_{i}}(g, V_{i}^{h}) is the determinant (in K_{i}) of the action of g in V_{i}^{h}. Since P is a pgroup, this determinant is in ⟨ζ_{r_{i}}⟩. Then Theorem 9.5 in [13] can be written in the following way.

2.1. Theorem. Let p be an odd prime and consider T and ψ_{h} : C_{P} (h) → T, for each h ∈ P, as before. Then

Cl_{1}(ℤP) ≅ T/⟨Imψ_{h} | h ∈ P ⟩.

Now, since p is odd, if we let S = {S_{1}, …, S_{k}} be a genetic basis of P, then N_{P} (S_{i})/S_{i} is cyclic for every 1 ⩽ i ⩽ k and each simple ℚP -module V_{i} is isomorphic to V (S_{i}) = Indinf_{N_{P} (S_{i})/S_{i}}^{P}Φ_{N_{P} (S_{i})/S_{i}}. Then the abelian group T defined before

is isomorphic to Γ(P) = ∏_{i=1}^{k} (N_{P} (S_{i})/S_{i}). This is because we can see the module

Φ_{N_{P} (S_{i})/S_{i}} ≅ ℚ(ζ_{r_{i}}), where p^{r_{i}} = |N_{P} (S_{i})/S_{i}| as actually being generated by a generator of N_{P} (S_{i})/S_{i} and thus the action of N_{P} (S_{i})/S_{i} on it can be seen as multiplication on the group. In particular, det_{K_{i}}(g, V_{i}^{h}), which is an element of the field K_{i}, can be regarded as an element in N_{P} (S_{i})/S_{i}. The first step in re-writing Theorem 2.1 is to find this element, for every S_{i}, every element h ∈ P and every g ∈ C_{P} (h).

2.2. Notation. Let p be and odd prime. If V is a simple ℚP -module and S is a genetic subgroup of P corresponding to V, we will write det_{N_{P} (S)/S}(g, V^{h}) for the element in N_{P} (S)/S corresponding to det_{K}(g, V^{h}), where K = End_{ℚP} (V).

2.3. Lemma. Suppose p is an odd prime. Let V be a simple ℚP -module and S be a genetic subgroup of P corresponding to V. Take an element h in P. Let H = ⟨h⟩ and [H\P/N_{P} (S)] be a set of representatives of the double cosets of P on H and N_{P} (S). Then, for g ∈ C_{P} (H), we have

det_{N_{P} (S)/S}(g, V^{h}) = ∏_{\begin{matrix}x∈[H\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} \overline{n_{g, x}},

where n_{g, x} is an element in N_{P} (S) given by the action σ_{g} of g in [H\P/N_{P} (S)], that is gx = h_{g, x}σ_{g}(x)n_{g, x}, with h_{g,x} ∈ H and σ_{g}(x) ∈ [H\P/N_{P} (S)] (even though the element n_{g, x} may not be unique, its class in N_{P} (S)/S is, thanks to the conditions on x). Proof. Set Φ for Φ_{N_{P} (S)/S}. Let [P/N_{P} (S)] be a set of representatives of the cosets of P in N_{P} (S). Since V ≅ Indinf_{N_{P} (S)/S}^{P}Φ, we can write it as ⨁_{a∈[P/N_{P} (S)]} a ⊗ Φ. The action of

y ∈ P is given by y(a ⊗ ω) = ya ⊗ ω, which is equal to τ_{y}(a) ⊗ \overline{n_{y, a}}ω, if ya = τ_{y}(a)n_{y, a} for a corresponding n_{y, a} in N_{P} (S), with \overline{n_{y, a}} being its class in N_{P} (S)/S.

If y ∈ H fixes an element u = ∑_{a∈[P/N_{P} (S)]} a ⊗ ω_{a} of V, we have

∑_{a∈[P/N_{P} (S)]} τ_{y}(a) ⊗ \overline{n_{y, a}}ω_{a} = ∑_{a∈[P/N_{P} (S)]} a ⊗ ω_{a} = ∑_{a∈[P/N_{P} (S)]} τ_{y}(a) ⊗ ω_{τ_{y}(a)}.

That is, for every a ∈ [P/N_{P} (S)] we should have \overline{n_{y, a}}ω_{a} = ω_{τ_{y}(a)}. If τ_{y}(a) = a, then we should have that y is in H ∩^{a}N_{P} (S) and that H^{a} ∩ N_{P} (S) ⩽ S, if ω_{a} is different from zero. We consider then the set [H\P/N_{P} (S)] and we have that

u = ∑_{\begin{matrix}x∈[H\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} ∑_{z∈[H/H∩^{x}N_{P} (S)]} zx ⊗ ω_{x}.

This means that a ℚ-basis for V^{h} is given by μ_{x, ω} = ∑_{z∈[H/H∩^{x}N_{P} (S)]} zx ⊗ ω with x

running over F = {x ∈ [H\P/N_{P} (S)] | H^{x} ∩ N_{P} (S) ⩽ S} and ω running over a ℚbasis of Φ. Since Φ ≅ ℚ(ζ_{r}) ≅End_{ℚP} (V), where ζ_{r} is a primitive p^{r}-th root of unity, if the order of N_{P} (S)/S is p^{r}, then a ℚ(ζ_{r})-basis of V^{h} is the set of μ_{x} = μ_{x, 1} with x running over F.

Now, for μ_{x} we have that gμ_{x} is equal to

∑_{z∈[H/H∩^{x}N_{P} (S)]} zh_{g, x}σ_{g}(x) ⊗ \overline{n_{g, x}} 1

if gx = h_{g, x}σ_{g}(x)n_{g, x}. That is

gμ_{x, 1} = h_{g, x}μ_{σ_{g}(x), \overline{n_{g, x}}} = μ_{σ_{g}(x), \overline{n_{g, x}}},

since μ_{σ_{g}(x), \overline{n_{g, x}}} is in V^{h}. So we can write gμ_{x} = \overline{n_{g, x}}μ_{σ_{g}(x)}, with \overline{n_{g, x}} seen as an element in the field ℚ(ζ_{r}). This implies that the action of g in V^{h} is given by a monomial matrix A, the coefficient in the non-zero entry of a row being \overline{n_{g,x}}. Then, the determinant of A is the product of the signature of the permutation σ_{g} by the product of the coefficients \overline{n_{g, x}}. Since p is odd, the signature is +1, and

det(A) = ∏_{\begin{matrix}x∈[H\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} \overline{n_{g, x}}.

Our final version of Theorem 2.1 is the following.

2.4. Theorem. Let p be an odd prime and P be a finite p-group. Take a set C of representatives of conjugacy classes of cyclic subgroups of P. For each H ∈ C, let \overline{E}_{H} be a generating set of the factor group C_{P} (H)/H and E_{H} ⊆ C_{P} (H) be a set of representatives of the classes gH ∈ \overline{E}_{H}. Let also S be a genetic basis of P and for each S ∈ S, let [H\P/N_{P} (S)] be a set of representatives of the double cosets of P on H and N_{P} (S). Then

Cl_{1}(ℤP) ≅ (∏_{S∈S} (N_{P} (S)/S)) /R,

where R is the subgroup generated by the elements u_{H,g} = (u_{H,g,S})_{S∈S}, for H ∈ C and g ∈ E_{H}, with

u_{H,g,S} = ∏_{\begin{matrix}x∈[H\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} \overline{n_{g, x}}

where n_{g, x} is an element in N_{P} (S) given by the action σ_{g} of g on [H\P/N_{P} (S)], that is gx = h_{g, x}σ_{g}(x)n_{g, x}, for h_{g,x} ∈ H and σ_{g}(x) ∈ [H\P/N_{P} (S)].

Proof. As we said at the beginning of the section, since p is odd, we have Cl_{1}(ℤP) ≅ Γ(P)/R, where

Γ(P) = ∏_{S∈S} (N_{P} (S)/S)

and R is the subgroup generated by all the elements u_{h,g} = (u_{h,g,S})_{S∈S}, with g ∈ C_{P} (h) and u_{h,g,S} = det_{N_{P} (S)/S} (g, V (S)^{h}), where V (S) = Indinf_{N_{P} (S)/S}^{P}Φ_{N_{P} (S)/S}.

We first observe that u_{h,g} = u_{^{y}h,^{y}g}, for any y ∈ G. Indeed, setting V = V (S), we have a commutative diagram

\xymatrix{V^{h} \ar[r]^{g} \ar[d]_{y} & V^{h} \ar[d]^{y} \\ V^{^{y}h} \ar[r]^{^{y}g} & V^{^{y}h}}

where the arrows are given by the actions of the labelling elements. It follows that the determinant of ^{y}g acting on V^{^{y}h} is equal to the determinant of g acting on V^{h}. Hence to generate the subgroup R of Γ(P), it suffices to take the elements u_{h,g}, where (h, g) runs through a set of representatives of conjugacy classes of pairs of commuting elements in P.

Now clearly for each h ∈ H, the map g ↦ u_{h,g} is a group homomorphism from C_{P} (h) to Γ(P), hence R is generated by the elements u_{h,g}, where h ∈ P and g runs through a set of generators of C_{P} (h). Moreover, as h acts as the identity on V^{h}, the group generated by h is contained in the kernel of this morphism.

Finally, by Lemma 2.3, setting H = ⟨h⟩, we have

u_{h,g,S} = ∏_{\begin{matrix}x∈[H\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} \overline{n_{g, x}}

and this depends only on H, so we may denote it by u_{H,g,S}, and by u_{H,g} the corresponding element of Γ(P).

It follows that R is generated by the elements u_{H,g}, where H is a cyclic subgroup of P up to conjugation, and for a given H, the element g runs through a subset of C_{P} (H) which, together with H, generates C_{P} (H). This completes the proof.

To finish the section, we prove that if N is a normal subgroup of a finite p-group P with p odd, then there is surjective deflation morphism

Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)).

2.5. Proposition. Let p be an odd prime and P be a finite p-group. Suppose N is a normal subgroup of P. Let S be a genetic basis of P and S_{N} = {S ∈ S | N ⩽ S}. Let B̃ be the set of subgroups S̃ = S/N of P̃ = P/N, for S ∈ S_{N}.

1. The set {S̃ | S ∈ S_{N}} is a genetic basis of P̃, and for S ∈ S_{N}, the projection P ↦ P̃ induces an isomorphism π_{S} : N_{P} (S)/S → NP̃(S̃)/S̃.

2. The composition

\xymatrix{s : Γ(P) = ∏_{S∈S} (N_{P} (S)/S) \ar[r] & ∏_{S∈S_{N}} (N_{P} (S)/S) \ar[r]^{∏ π_{S}} & ∏_{S̃∈S̃} (NP̃(S̃)/S̃) = Γ(P̃)}

induces a surjective deflation morphism Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)). In particular Cl_{1}(ℤ(P/N)) is isomorphic to a quotient of Cl_{1}(ℤP).

Proof. For Assertion 1, it is clear from the definitions that if S/N is a genetic subgroup of P/N, then S is a genetic subgroup of P. Moreover the irreducible representation of P associated to S is obtained by inflation from P/N to P of the irreducible representation of P/N associated to S/N, up to the obvious isomorphism π_{S} : N_{P} (S)/S ≅ NP̃(S̃)/S̃.

For Assertion 2, as the map s : Γ(P) → Γ(P̃) is surjective, all we have to check is that the subgroup R of defining relations for Cl_{1}(ℤP) ≅ Γ(P)/R is mapped by s inside the corresponding subgroup R̃ of defining relations for Cl_{1}(ℤP̃) ≅ Γ(P̃)/R̃. So let H be a cyclic subgroup of P, let g ∈ C_{P} (H), and let S ∈ S_{N}. Then H̃ = HN/N is cyclic, and g̃ = gN ∈ CP̃(H̃). Moreover the map

H\P/N_{P} (S) ∋ HxN_{P} (S) ↦ H̃x̃NP̃(S̃) ∈ H̃\P̃/NP̃(S̃),

where x̃ = xN ∈ P̃, is a bijection, since HxN_{P} (S) = HxNN_{P} (S) = HNxN_{P} (S) as N_{P} (S) ⩾ S ⩾ N. Hence we may identify the sets of representatives [H\P/N_{P} (S)] and [H̃\P̃/NP̃(S̃)] via this map. Moreover

H^{x} ∩ N_{P} (S) ⩽ S ⟺ (HN)^{x} ∩ N_{P} (S) ⩽ S ⟺ H̃x̃ ∩ NP̃(S̃) ⩽ S̃.

Now, for x ∈ [H\P/N_{P} (S)] such that H^{x}∩N_{P} (S) ⩽ S, the equality gx = h_{g,x}σ_{g}(x)n_{g,x},

where h_{g,x} ∈ H, σ_{g}(x) ∈ [H\P/N_{P} (S)], and n_{g,x} ∈ N_{P} (S), also reads g̃x̃ = h̃_{g,x}{ \atop σ_{g}(x)}ñ_{g,x}

in P/N. In other words π_{S}(\overline{n_{g,x}}) = \overline{n_{g̃,x̃}}, that is s(u_{H,g}) = u_{H̃,g̃}. This completes the proof.

2.6. Remark. It is shown in [4] (where Γ(P) is called the genome of P) that the map s is the deflation map in a structure of biset functor on Γ, but we will not need the corresponding additional operations of induction, restriction and inflation in this paper.

3. Computing some Whitehead groups

The examples we will consider in this section are all finite p-groups with p odd that satisfy the hypothesis of Theorem 8.10 in [13], so the group SK_{1}(ℤP) is equal to Cl_{1}(ℤP). Hence, we will use Theorem 2.4 to calculate Cl_{1}(ℤP). If P is abelian, Theorem 2.4 has a simpler expression, as it was already noted in Observation 1.13 of [14].

As we said in the introduction, to calculate the free rank of the Whitehead group of the groups in question, we will use Theorem 2.6 in [13]. We will also use Exercise 13.9 in Serre [16], which says that if G is a group of odd order and c is the number of irreducible non-isomorphic complex representations of G, then (c + 1)/2 is the number of irreducible non-isomorphic real representations of G.

We introduce some notation that will be helpful in both of our examples.

3.1. Notation. Let p be a prime. Suppose that W is a finite-dimensional vector space over the finite field 𝔽_{p}. We denote by S(W) the symmetric algebra of W and by S^{p}(W) its homogeneous part of degree p. If ψ : W → 𝔽_{p} is a linear functional, the map

w_{1} ⊗_{𝔽_{p}} … ⊗_{𝔽_{p}} w_{n} ∈ W^{⊗n} ↦ ψ(w_{1}) … ψ(w_{n}) ∈ 𝔽_{p}

induces a well defined linear functional S(W) → 𝔽_{p}, that we denote by A ↦ A(ψ).

The choice of a basis {x_{1}, …, x_{k}} of W over 𝔽_{p} yields a standard identification of S(W) with the polynomial ring 𝔽_{p}[x_{1}, …, x_{k}].

With such an identification, if A = A(x_{1}, …, x_{k}) ∈ S(W) and ψ is a linear form on W, then A(ψ) = A(ψ(x_{1}), …, ψ(x_{k})). In particular A(ψ) = 0 for all ψ if and only if the polynomial function associated to A is equal to zero, that is if A(r_{1}, …, r_{k}) = 0 for any (r_{1}, …, r_{k}) ∈ 𝔽_{p}^{k}: indeed since {x_{1}, …, x_{k}} is a basis of W, for any such ktuple (r_{1}, …, r_{k}) ∈ 𝔽_{p}^{k}, there is a unique linear form ψ on W such that ψ(x_{i}) = r_{i} for 1 ⩽ i ⩽ k.

3.2. Elementary abelian p-groups.

3.3. Lemma. Let p be an odd prime and P be an elementary abelian p-group of rank k, the free rank of W h(P) is equal to

\frac{(p^{k} − 1)(p − 3)}{2(p − 1)}.

Proof. The number of non-isomorphic irreducible ℝ-representations of P is (p^{k} +1)/2. On the other hand, the number of non-isomorphic irreducible ℚ-representations of P is equal to (p^{k} +p−2)/(p−1), since the genetic basis for P is given by all its subgroups of index p plus P itself. The result follows from Theorem 2.6 in [13].

The description of SK_{1} for elementary abelian groups appeared first in Alperin et al. [1], we prove this result now using genetic bases. Our proof has some similarities with the one appearing in [1], but it is a bit simpler, thanks to the use of genetic bases. The proof will also be useful when dealing with extra-special p-groups.

3.4. Lemma. Let p be a prime and let W be a finite-dimensional vector space over 𝔽_{p}. For x and y in W, we set B_{x, y} = x^{p−1}y ∈ S^{p}(W). Then S^{p}(W) is generated by the elements B_{x, y} with x and y running over W. Proof. Recall that if x_{1}, …, x_{m} are m (not necessarily different) commuting variables, then for any n

(x_{1} + ⋯ + x_{m})^{n} = ∑_{\begin{matrix}α_{1},...,α_{m} s.t. \\ α_{1}+···+α_{m}=n \end{matrix}} \frac{n!}{α_{1}! ⋯ α_{m}!}x_{1}^{α_{1}} ⋯ x_{m}^{α_{m}}.

This allows us to show that for any n

(3.5)  ∑_{∅≠A⊆{1,...,n}}(−1)^{n−|A|} (∑_{i∈A} x_{i})^{n} = n! x_{1} ⋯ x_{n},

and so if p is prime, then

∑_{∅≠A⊆{1,...,p−1}}(−1)^{p−1−|A|} (∑_{i∈A} x_{i})^{p−1} x_{p} = (p − 1)! x_{1} ⋯ x_{p},

which by Wilson’s Lemma gives

∑_{∅≠A⊆{1,...,p−1}}(−1)^{p−|A|} (∑_{i∈A} x_{i})^{p−1} x_{p} = x_{1} ⋯ x_{p},

that is

(3.6)  ∑_{∅≠A⊆{1,...,p−1}}(−1)^{p−|A|}B_{∑_{i∈A} x_{i},x_{p}} = x_{1} ⋯ x_{p}.

This completes the proof, by taking x_{1}, …, x_{p} ∈ W = S^{1}(W), which indeed commute in S(W).

3.7. Theorem. [Theorem 2.4 in [1]] Let p be an odd prime and P be an elementary abelian p-group of rank k, then SK_{1}(ℤP) is isomorphic to (C_{p})^{N} where

N = \frac{p^{k} − 1}{p − 1} − \binom{p + k − 1}{p}.

Proof. As we said before, the genetic basis of P consists of P itself and all its subgroups of index p, so SK_{1}(ℤP) is isomorphic to the quotient of

Γ(P) = ∏_{[P:Q]=p}(P/Q)

by the subgroup generated by the elements u_{x, y} for x, y in P, where

(u_{x, y})_{Q} = \{ \begin{matrix}yQ & if x ∈ Q \\ 1 & otherwise. \end{matrix}

On the other hand, we can see P as a vector space over 𝔽_{p}, and for each subgroup Q of index p, i.e. for each hyperplane Q of P, consider ψ_{Q} : P → 𝔽_{p}, a linear functional with kernel Q. The product of these ψ_{Q} induces an isomorphism from Γ(P) to

V = ∏_{[P:Q]=p} 𝔽_{p},

and the elements u_{x, y} can be seen as

(u_{x, y})_{Q} = \{ \begin{matrix}ψ_{Q}(y) & if x ∈ Q \\ 0 & otherwise. \end{matrix}

We define a morphism r : S^{p}(P) → V, sending A ∈ S^{p}(P) to the vector whose Qcomponent is equal to A(ψ_{Q}). We will show that Im(r) is equal to the subspace of V generated by the elements u_{x, y}, and that r is injective. This will give us the result.

We show first that the elements u_{x, y} are in Im(r). For x, y ∈ P, let B_{x, y} = x^{p−1}y ∈ S^{p}(P). Then

B_{x, y}(ψ_{Q}) = ψ_{Q}(x)^{p−1}ψ_{Q}(y) = \{ \begin{matrix}0 & if x ∈ Q \\ ψ_{Q}(y) & otherwise \end{matrix}

since λ^{p−1} = 1 if λ is in 𝔽_{p} − {0} and 0^{p−1} = 0. In particular B_{y,y}(ψ_{Q}) = ψ_{Q}(y) for all y ∈ P, thus r(B_{y,y} − B_{x, y}) = u_{x, y}.

On the other hand, by Lemma 3.4, S^{p}(P) is generated by the elements B_{x, y} where x and y run through P, so we have that Im(r) is contained in (hence equal to) the subspace of V generated by the elements u_{x, y}.

Finally, we prove that r is injective. Let A be in the kernel of r, then A(ψ_{Q}) = 0 for every Q of index p in P. If ψ is any other linear functional of P with kernel Q, then there exists λ ∈ 𝔽_{p} such that ψ = λψ_{Q}, and so A(ψ) = λ^{p}A(ψ_{Q}) = 0, since A is homogeneous of degree p. Choosing a basis of P over 𝔽_{p} as in Notation 3.1, we can view A as a homogeneous polynomial of degree p, and the polynomial function A is zero. It remains to see that A is actually the zero polynomial, but this will be clear after the following lemma.

3.8. Lemma. Let 𝔽 be a field of characteristic p > 0 and k be a positive integer. Let A be a homogeneous polynomial of degree d ⩽ p in k variables x_{1}, …, x_{k}. If A(x_{1}, …, x_{k}) = 0_{𝔽} for all (x_{1}, …, x_{k}) ∈ 𝔽^{k}, then A = 0. Proof. We proceed by induction on k. If k = 1, then A is of the form A(x_{1}) = ax_{1}^{d}, for some a ∈ 𝔽. Hence A(1_{𝔽}) = 0_{𝔽} = a, and A is the zero polynomial, starting induction.

Assume the results holds for homogeneous polynomials of degree d ⩽ p in less than k variables. Write

A(x_{1}, …, x_{k}) = A_{d}(x_{2}, …, x_{k})x_{1}^{d} + A_{d−1}(x_{2}, …, x_{k})x_{1}^{d−1} + … + A_{0}(x_{2}, …, x_{k}),

where A_{i}(x_{2}, …, x_{k}) is either the zero polynomial or a homogeneous polynomial of degree d − i in x_{2}, …, x_{k}, for 0 ⩽ i ⩽ d.

Let v = (x_{2}, …, x_{k}) be an element in 𝔽^{k−1}. For each i ∈ {0, … d}, set a_{i} = A(x_{2}, …, x_{k}). Then the polynomial

P_{v}(x_{1}) = a_{d}x_{1}^{d} + a_{d−1}x_{1}^{d−1} + … + a_{0}

vanishes for all x_{1} ∈ 𝔽_{p} ⊆ 𝔽. If d < p, then P_{v} must be the zero polynomial, because otherwise it would have more than d different roots. It follows that a_{i} = 0_{𝔽} for 0 ⩽ i ⩽ d. Since this argument holds for any v ∈ 𝔽^{k−1}, we have A_{i}(x_{2}, …, x_{k}) = 0_{𝔽} for any v = (x_{2}, …, x_{k}) ∈ 𝔽^{k−1}. If A_{i} ≠ 0 for some i ∈ {0, …, d}, then A_{i} is homogeneous of degree d − i ⩽ p, and it follows by induction that A_{i} = 0 for 0 ⩽ i ⩽ d, contradicting the assumption. Hence A_{i} = 0 for 0 ⩽ i ⩽ d, so A = 0.

Now if d = p, then there exists λ = λ(x_{2}, …, x_{k}) ∈ 𝔽 such that

a_{p}x_{1}^{p} + a_{p−1}x_{1}^{p−1} + … + a_{0} = λ(x_{1}^{p} − x_{1}).

Hence a_{i} = 0_{𝔽} = A_{i}(x_{2}, …, x_{k}) for i = 0 and for 2 ⩽ i ⩽ p − 1. Since this holds for any (x_{2}, …, x_{k}) ∈ 𝔽^{k−1}, by induction we have A_{i} = 0 for i = 0 and for 2 ⩽ i ⩽ p−1. Finally a_{1} = −λ = A_{1}(x_{2}, …, x_{k}) and a_{p} = λ = A_{p}(x_{2}, …, x_{k}) for any (x_{2}, …, x_{k}) ∈ 𝔽^{k−1}, that is

A_{1}(x_{2}, …, x_{k}) = −A_{p}(x_{2}, …, x_{k}).

But A_{p} is homogeneous of degree 0, i.e. it is a constant polynomial. Taking x_{2} = x_{3} = … = x_{k} = 0_{𝔽} in the previous equality, we get

A_{1}(0_{𝔽}, …, 0_{𝔽}) = 0_{𝔽} = A_{p}(0_{𝔽}, …, 0_{𝔽}),

where the first equality holds because A_{1} is homogeneous of degree p−1 > 0. It follows that A_{p} = 0, and now A_{1}(x_{2}, …, x_{k}) = 0_{𝔽} for any (x_{2}, …, x_{k}) ∈ 𝔽^{k−1}. By induction A_{1} = 0, hence A = 0.

3.9. Corollary. Let p be an odd prime, and P be a finite p-group. If |P/Φ(P)| = p^{k}, then SK_{1}(ℤP) has a subquotient isomorphic to (C_{p})^{N}, where

N = \frac{p^{k} − 1}{p − 1} − \binom{p + k − 1}{p},

and in particular SK_{1}(ℤP) ≠ 0 if k ⩾ 3.

Proof. Indeed Cl_{1}(ℤ(P/Φ(P))) ≅ (C_{p})^{N}, for N = \frac{p^{k}−1}{p−1} − \binom{p+k−1}{p}, by Theorem 3.7. Moreover, this group is a quotient of Cl_{1}(ℤP), by Proposition 2.5. Finally Cl_{1}(ℤP) is a subgroup of SK_{1}(ℤP), and N > 0 if k ⩾ 3.

3.10. Extra-special and almost extra-special p-groups.

We begin by finding a genetic basis of an (almost) extra-special p-group.

3.11. Proposition. Let p be a prime and P be an (almost) extra-special p-group. A genetic basis of P is given by all its subgroups of index p, together with P and a subgroup Y of maximal order such that Y ∩ Z(P) = 1. In particular P has a unique faithful rational irreducible representation, up to isomorphism. Proof. We abbreviate Z(P) by Z.

By theorems 1.7 and 1.8, the subgroups of P of index 1 or p are genetic and are not linked modulo _{P} . They clearly intersect Z non-trivially. On the other hand, any genetic subgroup S ≠ P which intersects Z non-trivially must have index p, since P′ ⩽ S and so the cyclic group P/S ≅ (P/P′)/(S/P′), should have order p. This implies that if there is another group in S, it must intersect Z trivially.

Let Y ⩽ P be of maximal order with the property Y ∩ Z = 1. By Lemma 1.18, we have that C_{P} (Y) = N_{P} (Y) = Y Z. In particular N_{P} (Y) ⊴ P and N_{P} (Y)/Y ≅ Z is a Roquette group. Then Y is genetic, by Remark 1.12. Also, by Lemma 1.18, we have that if Y_{1} ⩽ P is a group such that Y_{1} ∩ Z = 1, but it is not maximal order with this property, then C_{P} (Y_{1}) = N_{P} (Y_{1}), but N_{P} (Y_{1})/Y_{1} is not cyclic. Thus Y_{1} is not a genetic subgroup of P.

Finally, if Y is a subgroup of P of maximal order such that Y ∩ Z = 1, then Y has |P/Y Z| = |Y | distinct conjugates in P, by Lemma 1.18. These conjugates are subgroups of index p in the elementary abelian group Y P′, and they all intersect trivially (that is, they don’t contain) the group P′. Since there are exactly |Y | subgroups of Y P′ not containing P′, these subgroups are exactly the conjugates of Y in P.

Now if Y_{0} is another subgroup of P such that Y_{0} ∩ Z = 1, then Y_{0} ∩ Y P′ is a subgroup of Y P′ which does not contain P′. Hence it is contained in some conjugate of Y, and there exists x ∈ P such that Y_{0} ∩ Y P′ ⩽ Y^{x}. It follows that Y_{0} ∩ Y Z ⩽ Y^{x}, for Y P′ is the subgroup of Y Z consisting of elements of order at most p. In other words

xY_{0} ∩ Z_{P} (Y) = ^{x}Y_{0} ∩ Y Z = ^{x}(Y_{0} ∩ Y Z) ⩽ Y.

Now if Y_{0} is another subgroup of maximal order such that Y_{0} ∩ Z = 1, exchanging the roles of Y and Y_{0} in the previous argument shows that there also exists an element y ∈ P such that ^{y}Y ∩ Y_{0}Z ⩽ Y_{0}. By Theorem 1.8, it follows that Y_{0} _{P} Y. The last assertion now follows from Lemma 1.10.

As a first consequence of this result we have.

3.12. Lemma. Let p be an odd prime, and n be a positive integer.

1. Let P be an extra-special p-group of order p^{2n+1}. Then the free rank of W h(P) is equal to

\frac{(p^{2n} + p − 2)(p − 3)}{2(p − 1)}.

2. Let P be an almost extra-special group of order p^{2n+2}. Then the free rank of W h(P) is equal to

\frac{(p^{2n+1} + p^{2} + p + 1)(p − 3) + 8}{2(p − 1)}.

Proof. The free rank of W h(P) is equal to r − q, where r (resp. q) is the number of irreducible real (resp. rational) representations of P, up to isomorphism. In general, for a finite p-group P and a field K of characteristic 0, the irreducible representations of P over K can be recovered from the knowledge of a genetic basis B of P (see [2]): this is because the functor R_{K} of representations of p-groups over K is rational in the sense of Definition 10.1.3 of [5], as can be easily deduced from Theorem 10.6.1 of [5]. A proof of this fact can also be found in [6]. In particular, the number l_{K}(P) of such representations, up to isomorphism, is equal to

l_{K}(P) = ∑_{S∈B} ∂l_{K}(N_{P} (S)/S),

where ∂l_{K}(Q) denotes the number of faithful irreducible representations of a p-group Q over K, up to isomorphism. For a Roquette p-group Q, we have moreover ∂l_{K}(Q) = 1 if Q = 1, and ∂l_{K}(Q) = l_{K}(Q) − l_{K}(Q/Z) otherwise, where Z is the unique central subgroup of order p of Q.

If p is odd, all the groups N_{P} (S)/S, for S ∈ B, are cyclic. Now for Q = C_{p^{m}}, with m ⩾ 0, we have

l_{ℝ}(Q) = \frac{p^{m} + 1}{2} and l_{ℚ}(Q) = m + 1.

It follows that ∂l_{ℝ}(Q) = \frac{p^{m}−p^{m−1}}{2} if m > 0, and ∂l_{ℝ}(Q) = 1 if m = 0. On the other hand ∂l_{ℚ}(Q) = 1 for any m.

In case P is extra-special of order p^{2n+1}, the genetic basis obtained in Proposition 3.11 consists of the group S = P, for which N_{P} (S)/S is trivial, of \frac{p^{2n}−1}{p−1} subgroups S of index p in P, for which N_{P} (S)/S ≅ C_{p}, and the subgroup S = Y, for which N_{P} (S)/S ≅ Z(P) ≅ C_{p}. This gives

r = l_{ℝ}(P) = 1 + \frac{p^{2n} − 1}{p − 1} \frac{p − 1}{2} + \frac{p − 1}{2} = \frac{p^{2n} + p}{2},

and

q = l_{ℚ}(P) = 1 + \frac{p^{2n} − 1}{p − 1} + 1 = \frac{p^{2n} + 2p − 3}{p − 1}.

In case P is almost extra-special of order p^{2n+2}, the genetic basis obtained in Proposition 3.11 consists of the group S = P, for which N_{P} (S)/S is trivial, of \frac{p^{2n+1}−1}{p−1} subgroups S of index p in P, for which N_{P} (S)/S ≅ C_{p}, and the subgroup S = Y, for which N_{P} (S)/S ≅ Z(P) ≅ C_{p^{2}}. This gives

r = l_{ℝ}(P) = 1 + \frac{p^{2n+1} − 1}{p − 1} \frac{p − 1}{2} + \frac{p^{2} − p}{2} = \frac{p^{2n+1} + p^{2} − p + 1}{2},

and

q = l_{ℚ}(P) = 1 + \frac{p^{2n+1} − 1}{p − 1} + 1 = \frac{p^{2n+1} + 2p − 3}{p − 1}.

This completes the proof.

To calculate Cl_{1}(ℤP) we will need the following result.

3.13. Lemma. Let p be an odd prime. Let W be a vector space over 𝔽_{p} of finite dimension k, which is endowed with a bilinear, alternating form b : W × W → 𝔽_{p}. Suppose that the rank of b is not equal to 2.

For x and y in W, we still set B_{x, y} = x^{p−1}y ∈ S^{p}(W). Then we have

S^{p}(W) = ⟨B_{x, y} | x, y ∈ W s.t. b(x, y) = 0⟩.

Proof. We will write O for ⟨B_{x, y} | x, y ∈ W s.t. b(x, y) = 0⟩. We will prove that B_{x,y} ∈ O for any x, y ∈ W, and by Lemma 3.4, it will follow that S^{p}(W) = O.

Observe that, since B_{x,y} ∈ O for an x, y ∈ W with b(x, y) = 0, it follows from formula 3.6 that x_{1}x_{2} … x_{p−1}y ∈ O for any elements x_{1}, …, x_{p−1} of W such that b(x_{i}, y) = 0 for 1 ⩽ i ⩽ p − 1.

If b = 0, there is nothing to prove. Otherwise let x, y ∈ W such that b(x, y) ≠ 0. Since B_{x,λy} = λB_{x,y} for λ ∈ 𝔽_{p}, to prove that B_{x,y} ∈ O, we can assume without loss of generality that b(x, y) = 1, up to replacing y by a suitable scalar multiple. If the restriction of b to ⟨x, y⟩^{⊥} was identically 0, then ⟨x, y⟩^{⊥} would be precisely the radical of b, so b would have rank 2, contradicting our assumption. Hence we can find z, t ∈ ⟨x, y⟩^{⊥} such that b(z, t) ≠ 0, and up to replacing t by some scalar multiple, we can assume that b(z, t) = 1.

Now let α ∈ 𝔽_{p}, and set u = αx + t and v = y + αz. Then

b(u, v) = αb(x, y) + α^{2}b(x, z) + b(t, y) + αb(t, z) = 0,

since b(x, y) = 1 = −b(t, z) and b(x, z) = 0 = b(t, y).

It follows that B_{u,v} ∈ O. But B_{u,v} is equal to

(3.14) (αx + t)^{p−1}(y + αz) = ∑_{i=0}^{p−1} \binom{p − 1}{i}α^{i}x^{i}yt^{p−1−i} + ∑_{i=0}^{p−1} \binom{p − 1}{i}α^{i+1}t^{p−1−i}zx^{i}.

By the observation at the beginning of the proof, the element x^{i}yt^{p−1−i} is in O whenever p − 1 − i > 0, since b(x, t) = b(y, t) = b(t, t) = 0. Similarly t^{p−1−i}zx^{i} ∈ O if i > 0, since b(t, x) = b(z, x) = b(x, x) = 0. It follows that in (3.14), the only elements possibly not in O correspond to p − 1 − i = 0 in the first summation and to i = 0 in the second. Hence

α^{p−1}x^{p−1}y + αt^{p−1}z ∈ O,

and this holds for any α ∈ 𝔽_{p}. For α = 1, this gives x^{p−1}y +t^{p−1}z ∈ O, and for α = −1, this gives x^{p−1}y − t^{p−1}z ∈ O. It follows that x^{p−1}y ∈ O, as was to be shown.

3.15. Remark. If the rank of b is equal to 2, then the result of Lemma 3.13 is no longer true: for example in the non-degenerate case, that is when W has dimension 2, saying that b(x, y) = 0 for x ≠ 0 is equivalent to saying that y is a scalar multiple of x. In this case O is the subspace of S^{p}(W) generated by the elements x^{p}, for x ∈ W. So O has dimension 2, whereas S^{p}(W) has dimension p + 1.

We now come to our main theorem, describing the structure of Cl_{1}(ℤP) when P is an extra-special or almost extra-special p-group for p odd. We first recall that Oliver ([13] Example 7 page 16) showed that if P is extra-special of order p^{3}, then Cl_{1}(ℤP) ≅ (C_{p})^{p−1}, and that if P is almost extra-special of order p^{4}, then Cl_{1}(ℤP) ≅ (C_{p})^{(p^{2}+p−2)/2}. Hence it what follows, we may assume that P is an extra-special group of order at least p^{5}, or an almost extra-special p-group of order at least p^{6}.

3.16. Notation. Let p be an odd prime and n be a positive integer. Let P be an extra-special p-group of order p^{2n+1} or an almost extra-special p-group of order p^{2n+2}. Let Z denote the center of P, and N = P′ ⩽ Z be the Frattini subgroup of P. Let Y be a subgroup of P of maximal order such that Y ∩ Z = 1, as in Proposition 3.11. Recall that Y is elementary abelian. In any case, the group P can be written as a semidirect product P = X · Y, where X ⩾ Z is an abelian normal subgroup of P with Y ∩ X = 1:

• If P is extra-special of exponent p, the group X is equal to C × X_{0}, for some subgroup X_{0} ≅ (C_{p})^{n} and C = N = Z.

• If P is extra-special of exponent p^{2}, the group X is equal to C × X_{0}, for some subgroup X_{0} ≅ (C_{p})^{n−1}, some subgroup C ≅ C_{p^{2}}, and N = Z < C.

• If P is almost extra-special, then X = C × X_{0}, for some subgroup X_{0} ≅ (C_{p})^{n}, and N < Z = C ≅ C_{p^{2}}.

So in all cases we have X = C × X_{0}, for some cyclic subgroup C ⩾ Z ⩾ N.

Moreover the subgroup Y is elementary abelian of order p^{n}. It is maximal subject to the condition Y ∩ Z = 1, so by Proposition 3.11, we have a genetic basis of P consisting of P itself, its subgroups of index p, and Y. The normalizer N_{P} (Y) is equal to Z · Y, so

Γ(P) ≅ ( ∏_{[P:Q]=p}(P/Q)) × Z.

We will often use an additive notation for Γ(P) and its components.

3.17. Theorem. Let p be an odd prime, and let P be an extra-special p-group of order at least p^{5}, or an almost extra-special p-group of order at least p^{6}. Let N = P′ be the Frattini subgroup of P, and Z be the center of P. Then there is a split sequence of abelian groups

\xymatrix{0 \ar[r] & K \ar[r] & Cl_{1}(ℤP) \ar[r]^{^{Def{P \atop P/N}}} & Cl_{1}(ℤ(P/N)) \ar[r] & 0,}

where K is cyclic, generated by the image in Cl_{1}(ℤP) of the unique faithful rational irreducible representation of P. Moreover K is isomorphic to the quotient of Z by the subgroup generated by all the elements u_{H,g,Y} ∈ N_{P} (Y)/Y ≅ Z introduced in Theorem 2.4, where H is a cyclic subgroup of P and g ∈ C_{P} (H). Proof. The product ∏_{[P:Q]=p}(P/Q) identifies with Γ(P/N), and we have a surjective

projection map Def_{P/N}^{P} : Γ(P) → Γ(P/N), with kernel isomorphic to Z. By Proposition 2.5, this map induces a surjective deflation map

Def_{P/N}^{P} : Cl_{1}(ℤP) = Γ(P)/R → Cl_{1}(ℤ(P/N)) = Γ(P/N)/\overline{R},

where R is the subgroup of Γ(P) generated by the elements u_{H,g} introduced in Theorem 2.4, and H is a cyclic subgroup of P with g ∈ C_{P} (H). Similarly \overline{R} is the corresponding subgroup of Γ(P/N) generated by the elements u_{F,c}, where F is a cyclic subgroup of P/N and c ∈ P/N (we always have c ∈ C_{P/N}(F), as P/N is abelian).

The proof of Theorem 2.5 shows that Def_{P/N}^{P}(u_{H,g}) = u_{HN/N,gN}. Conversely, if F is a cyclic subgroup of P/N, generated by f, and if c ∈ P/N, then there exists a pair (H, g) of a cyclic subgroup H of P and an element g ∈ C_{P} (H) such that HN/N = F and gN = c if and only if b(f, c) = 0, where b is the bilinear alternating form on P/N with values in 𝔽_{p} induced by taking commutators in P. Our assumptions on P imply that the rank of b is not 2, so we can apply Lemma 3.13. This shows that the subspace of Γ(P/N) generated by the elements u_{F,c}, where F = ⟨f⟩ for f ∈ P/N, and c ∈ P/N such that b(f, c) = 0, generate S^{p}(P/N) = \overline{R}, by Lemma 3.4. It follows that Def_{P/N}^{P} induces a surjective map R → \overline{R}. Let L denote the kernel of this map. We have a commutative diagram with exact rows

\xymatrix{0 \ar[r] & L \ar[d]^{l} \ar[r] & R \ar[d]^{i} \ar[r]^{^{Def{P \atop P/N}}} & \overline{R} \ar[d]^{j} \ar[r] & 0 \\ 0 \ar[r] & Z \ar[r] & Γ(P) \ar[r]^{_{Def{P \atop P/N}}} & Γ(P/N) \ar[r] & 0}

where the vertical maps i and j are the inclusion maps. The Snake’s Lemma now shows that the map l is injective, and moreover we have an exact sequence of cokernels

(3.18)  \xymatrix{0 \ar[r] & K \ar[r] & Cl_{1}(ℤP) \ar[r]^{^{Def{P \atop P/N}}} & Cl_{1}(ℤ(P/N)) \ar[r] & 0,}

where K = Z/l(L). Since the kernel Z of Def_{P/N}^{P} corresponds to the component of Γ(P) indexed by Y, the image l(L) is generated by the components u_{H,g,Y} ∈ N_{P} (Y)/Y ≅ Z, where H is a cyclic subgroup of P and g ∈ C_{P} (H).

It remains to see that the exact sequence 3.18 splits. To see this, consider the completed diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & L \ar[d]^{l} \ar[r] & R \ar[d]^{i} \ar[r] & \overline{R} \ar[d]^{j} \ar[r] & 0 \\ 0 \ar[r] & Z \ar[d] \ar[r] & Γ(P) \ar[d]^{a} \ar[r]^{c} & Γ(P/N) \ar@{-->}[l]^{t} \ar[d]^{b} \ar[r] & 0 \\ 0 \ar[r] & K \ar[d] \ar[r] & Cl_{1}(ℤP) \ar[d] \ar[r]^{d} & Cl_{1}(ℤ(P/N)) \ar@{-->}[u]^{s} \ar[d] \ar[r] & 0, \\  & 0 & 0 & 0}

where a and b are the projection maps, and c, d are the respective deflation maps. The map b is split surjective, because Γ(P/N) and Cl_{1}(ℤ(P/N)) are both elementary abelian. Let s be a section of b. Similarly, the map c is split surjective by construction. Let t : Γ(P/N) → Γ(P) be a section of c. Then

d ∘ a ∘ t ∘ s = b ∘ c ∘ t ∘ s = b ∘ s = Id,

so the map a ∘ t ∘ s is a section of d. This completes the proof.

So to go further, we have to focus on the computation of the elements u_{H,g,Y} appearing in Theorem 3.17. First a technical lemma:

3.19. Lemma. Keeping Notation 3.16, let H be a cyclic subgroup of P. Then there exists a set D (possibly empty) of representatives of those double cosets HxN_{P} (Y) in P for which H^{x} ∩ N_{P} (Y) ⩽ Y, and a subgroup X_{1} of X_{0} of index at most p, such that DX_{1} = D. Proof. If there is no element x ∈ P such that H^{x} ∩ N_{P} (Y) ⩽ Y, then we take D = ∅ and X_{1} = X_{0}. So assume that H^{x} ∩ N_{P} (Y) ⩽ Y for some x ∈ P.

Since the derived group P′ = N is central in P, by standard formulae (see e.g. [10], Chapter 2, Lemma 2.2), for any two elements a, b ∈ P and any integer m, we have (ab)^{m} = a^{m}b^{m}[b, a]^{\binom{m}{2}}. Suppose that H is generated by h = ξη, with ξ ∈ X and η ∈ Y. We can write further ξ = γξ_{0}, for γ ∈ C and ξ_{0} ∈ X_{0}. We have H^{x} ∩ N_{P} (Y) = (H ∩ ZY)^{x} ⩽ Y. But h^{p} = ξ^{p}η^{p}[ξ, η]^{\binom{p}{2}} = ξ^{p} = γ^{p} since P′, X_{0} and Y are elementary abelian and \binom{p}{2} is a multiple of p. Moreover γ ∈ C, hence γ^{p} ∈ Z, as Z has index 1 or p in C. Thus γ^{p} ∈ H ∩ Z = (H ∩ Z)^{x} ⩽ (H ∩ ZY)^{x} ⩽ Y. It follows that γ^{p} = 1, and γ ∈ N ⩽ Z. Moreover h^{p} = 1.

If h = 1, then H\P/N_{P} (Y) = XY/ZY ≅ X/Z = (C/Z) × X_{0}, so we can take D = E × X_{0}, for a chosen set E = [C/Z] of Z-cosets in C, and X_{1} = X_{0}. Hence we can assume that h has order p, that is |H| = p.

Then there are two cases:

Case 1: Suppose that H ∩ N_{P} (Y) = 1, or equivalently H ≰ ZY. Then H^{g} ≰ ZY, for any g ∈ P, so H^{g} ∩ ZY = 1 ⩽ Y. In this case, we take a subgroup X_{1} of X_{0} such that X_{0} = ⟨ξ_{0}⟩ × X_{1}. Observe that X_{1} has index at most p in X_{0}. We take moreover a set E = [C/Z] of representatives of the cosets C/Z. We claim that the set D = EX_{1} has the required properties. Since obviously DX_{1} = D, all we have to show is that D is indeed a set of representatives of (H, N_{P} (Y))-double cosets in P.

So let e, e′ ∈ E and x_{1}, x′_{1} ∈ X_{1}, and assume that there exists an integer m, an element z ∈ Z and an element y ∈ Y, such that

e′x′_{1} = (h^{m})ex_{1}(zy).

Since h = γξ_{0}η, this also reads

e′x′_{1} = (γξ_{0}η)^{m}ex_{1}(zy) = (γξ_{0})^{m}η^{m}[η, γξ_{0}]^{\binom{m}{2}}ex_{1}zy
= γ^{m}ξ_{0}^{m}[η, γξ_{0}]^{\binom{m}{2}}ex_{1}η^{m}[η^{m}, ex_{1}]zy
e′x′_{1} = \underbrace{(γ^{m}[η, γξ_{0}]^{\binom{m}{2}}e[η^{m}, ex_{1}]z)}_{∈C} x_{1}(η^{m}y).

The left hand side is in X. The terms on the right hand side are all in X, except η^{m}y, which is in Y. Hence η^{m}y = 1, and

e′x′_{1} = \underbrace{(γ^{m}[η, γξ_{0}]^{\binom{m}{2}}e[η^{m}, ex_{1}]z)}_{∈C} x_{1}.

Since e′ ∈ C, x′_{1} ∈ X_{0}, x_{1} ∈ X_{0}, and since X = C × X_{0}, it follows that x′_{1} = x_{1} and e′ = (γ^{m}[η, γξ_{0}]^{\binom{m}{2}}e[η^{m}, ex_{1}]z). Since γ ∈ Z, this in turn implies e′ ∈ eZ, hence e′ = e. So the double cosets HdN_{P} (Y), for d ∈ D = EX_{1}, are all distinct.

It remains to show that any (H, N_{P} (Y))-double coset in P is of this form. So let g ∈ P = (C × ⟨ξ_{0}⟩ × X_{1}) · Y. Hence there exist c ∈ C, m ∈ ℕ, x_{1} ∈ X_{1}, and y ∈ Y such that g = cξ_{0}^{m}x_{1}y. There also exists e ∈ E and z′ ∈ Z such that c = ez′. It follows that

HgN_{P} (Y) = HgZY = Hcξ_{0}^{m}x_{1}yZY = Hcξ_{0}^{m}x_{1}ZY
= Hc(ξ_{0}η)^{m}[η, ξ_{0}]^{−\binom{m}{2}}η^{−m}x_{1}ZY
= Hch^{m}η^{−m}x_{1}ZY = Hcx_{1}η^{−m}[η^{−m}, x_{1}]ZY
= Hcx_{1}ZY = Hez′x_{1}ZY = Hex_{1}ZY,

as was to be shown, since ex_{1} ∈ D.

Case 2: Suppose now that H ⩽ ZY and H^{x} ⩽ Y. If D is a set of representatives of the (H^{x}, ZY)-double cosets in P, then xD is a set of representatives of the (H, ZY)- double cosets in P. So without loss of generality, we can assume that x = 1, that is H ⩽ Y. If g ∈ G, then HgZY = HZY g = ZY g = gZY, so H\P/ZY = P/ZY. If moreover H^{g} ⩽ Y, then as h^{g} = h[h, g], we have [h, g] ∈ Z ∩ Y = 1, and g ∈ C_{P} (H). Hence we seek for a set D of representatives of C_{P} (H)/ZY. But Y ⩽ C_{P} (H) ⩽ XY, so C_{P} (H) = C_{X}(H)Y, hence C_{P} (H)/ZY ≅ C_{X}(H)/Z.

Now C_{P} (H) is a subgroup of index p of P, by Lemma 1.18, since H has order p and H is not central in P. Hence C_{X}(H) has index p in X = C × X_{0}. So the group X_{1} = C_{X_{0}}(H) has index at most p in X_{0}, and moreover there exists a subgroup X_{2} ⩾ Z of C_{X}(H) such that C_{X}(H) = X_{2} × X_{1}. Choosing a set E of representatives of X_{2}/Z, the set D = EX_{1} is a set of representatives of C_{X}(H)/Z with the required properties.

This completes the proof.

3.20. Theorem. Let p be an odd prime, and let P be an extra-special p-group of order at least p^{5} or an almost extra-special p-group of order at least p^{6}. Set N = Φ(P) = P′.

1. The group Cl_{1}(ℤP) is isomorphic to K × (C_{p})^{M}, where M = \frac{p^{k−1}−1}{p−1} − \binom{p+k−2}{p} if |P | = p^{k}, and K is the kernel of Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)).

2. Moreover, the group K is:

(a) trivial if P is extra-special of order p^{5} and exponent p^{2}.

(b) of order p if P is extra-special of exponent p, or extra-special of exponent p^{2} and order at least p^{7}, or almost extra-special of order p^{6}.

(c) cyclic of order p^{2} if P is almost extra-special of order at least p^{8}.

Proof. We keep Notation 3.16 throughout. We know from Theorem 3.17 that Cl_{1}(ℤP) ≅ K × Cl_{1}(ℤ(P/N)). Moreover Cl_{1}(ℤ(P/N)) is elementary abelian of rank k − 1, so Assertion 1 follows from Theorem 3.7.

Moreover, the group K is isomorphic to the quotient of Z by the subgroup generated by the elements u_{H,g,Y}, where H is a cyclic subgroup of P, and g ∈ C_{G}(H). For such a pair (H, g),

u_{H,g,Y} = ∏_{x∈D} \overline{n_{g, x}}

where D is a chosen set of representatives of those double cosets HxN_{P} (Y) in P for which H^{x} ∩N_{P} (Y) ⩽ Y. For each x ∈ D, the element n_{g,x} ∈ N_{P} (Y) is chosen such that gx = h_{g,x}σ_{g}(x)n_{g,x}, where σ_{g} is the permutation of D induced by left multiplication by g ∈ C_{P} (H), and h_{g,x} ∈ H. The image \overline{n_{g,x}} of n_{g,x} in N_{P} (Y)/Y is well defined by this equality, thanks to the condition H^{x} ∩ N_{P} (Y) ⩽ Y.

Suppose that u_{H,g,Y} is non zero. We can apply Lemma 3.19: there is a set D of representatives of those double cosets HxN_{P} (Y) in P for which H^{x} ∩ N_{P} (Y) ⩽ Y, and a subgroup X_{1} of X_{0} of index at most p such that DX_{1} = D. Then for x ∈ D, we have gx = h_{g,x}σ_{g}(x)n_{g,x}, and for x_{1} ∈ X_{1}

gxx_{1} = h_{g,x}σ_{g}(x)n_{g,x}x_{1} = h_{g,x}σ_{g}(x)x_{1}n_{g,x}[n_{g,x}, x_{1}],

and n_{g,x}[n_{g,x}, x_{1}] ∈ N_{P} (Y) = ZY. In other words we have σ_{g}(xx_{1}) = σ_{g}(x)x_{1} and we can take n_{g,xx_{1}} = n_{g,x}[n_{g,x}, x_{1}]. Choosing a set E of representatives of cosets D/X_{1}, this gives

u_{H,g,Y} = ∏_{\begin{matrix}x∈E \\ x_{1}∈X_{1} \end{matrix}} \overline{n_{g,xx_{1}}} = ∏_{\begin{matrix}x∈E \\ x_{1}∈X_{1} \end{matrix}} \overline{n_{g,x}[n_{g,x}, x_{1}]}

= ∏_{x∈E} ∏_{x_{1}∈X_{1}} \overline{n_{g,x}[n_{g,x}, x_{1}]}
= ( ∏_{x∈E} \overline{n_{g,x}}^{|X_{1}|})( ∏_{x∈E} \overline{[n_{g,x}, ∏_{x_{1}∈X_{1}} x_{1}]} ).

Now ∏_{x_{1}∈X_{1}} x_{1} = 1, since X_{1} has odd order. Hence

(3.21)  u_{H,g,Y} = ∏_{x∈E} \overline{n_{g,x}}^{|X_{1}|}.

If |X_{1}| ⩾ |Z|, since \overline{n_{g,x}} ∈ Z for any x ∈ E, we then have u_{H,g,Y} = 0 in additive notation. This is the case if • either |Z| = p and |X_{0}| ⩾ p^{2}, i.e. if P is extra-special of exponent p and order at

least p^{5}, or extra-special of exponent p^{2} and order at least p^{7},

• or if |Z| = p^{2} and |X_{0}| ⩾ p^{3}, i.e. if P is almost extra-special of order at least p^{8}.

In each of these cases, we have u_{H,g,Y} = 0 for any cyclic subgroup H of P and any g ∈ C_{P} (H). It follows that K = Z in this case, as was to be shown.

So we are left with the special cases where P is either extra-special of order p^{5} and exponent p^{2}, or almost extra-special of order p^{6}.

Suppose first that P is extra-special of order p^{5} and exponent p^{2}. Then we have X = C × X_{0}, where C ≅ C_{p^{2}} and X_{0} ≅ C_{p}. On the other hand Y ≅ (C_{p})^{2} in this case. We take H = X_{0}, and we take for g a generator of C, so indeed g ∈ C_{P} (H). Now H ≰ ZY, so H^{x} ∩ ZY = 1 ⩽ Y for any x in P. Moreover H\P/ZY = H\XY/ZY ≅ H\X/Z ≅ C/Z, so we can take a set D of representatives of these double cosets consisting of a set of representatives of C/Z, e.g. D = {1, g, g^{2}, …, g^{p−1}}. Now for x = g^{i} with 0 ⩽ i ⩽ p − 2, we have gx = g^{i+1} ∈ D, so n_{g,x} = 1. But for x = g^{p−1}, we have gx = g^{p} = 1 · g^{p}, so n_{g,x} = g^{p} in this case. It follows that u_{H,g,Y} is equal to g^{p}, which is a generator of Z. Hence K is trivial in this case, as was to be shown. In this case Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)) is an isomorphism.

Suppose now that P is almost extra-special of order p^{6}. Then we have C = Z ≅ C_{p^{2}} and X_{0} ≅ (C_{p})^{2}. In this case, Equation 3.21 shows that u_{H,g,Y} is equal to v_{H,g}^{|X_{1}|} for some v_{H,g} ∈ Z, for any cyclic subgroup H of P and any g ∈ C_{P} (H). Moreover X_{1} has at least order p, since X_{0} has order p^{2}. It follows in additive notation that u_{H,g,Y} ∈ pZ.

But on the other hand, write X_{0} = A × B, where A and B have order p. If we take H = B, and for g a generator of Z, then g ∈ C_{P} (H). Moreover H ≰ ZY, so H^{x} ∩ ZY = 1 ⩽ Y for any x in P. Furthermore H\P/ZY = X/ZH ≅ A, so A is a set of representatives of the double cosets H\P/ZY. But for a ∈ A, we have ga = ag, so n_{g,a} = g ∈ Z. It follows that u_{H,g,Y} = g^{|A|} = g^{p}. Hence the subgroup of Z generated by all the elements u_{H,g,Y} is equal to pZ, so K ≅ Z/pZ ≅ C_{p} in this case, as was to be shown.

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Serge Bouc, CNRS-LAMFA, 33 rue St Leu, 80039, Amiens, France.

serge.bouc@u-picardie.fr

Nadia Romero, DEMAT, UGTO, Jalisco s/n, Mineral de Valenciana, 36240, Guanajuato, Gto., Mexico.

nadia.romero@ugto.mx