EVENS NORM AND RESTRICTION MAPS

IN mod-ppp COHOHOMOLOGY OF ppp-GROUPS

Pham Anh Minh

30 November 1998

1. Statements of main results

Let p be a prime number. Given a p-group G, denote by H^{∗}(G) the mod-p cohomology algebra of G. For every subgroup K of G, let Res_{K}^{G} be the restriction maps from H^{∗}(G) to H^{∗}(K). Consider the case where K = Keru, with 0 ≠ u ∈ H^{1}(G) = Hom(G, 𝔽_{p}). It was known that, for p = 2, KerRes_{K}^{G} is the principal ideal (u) generated by u. However, for p > 2, a corresponding statement does not hold: while a result of Quillen-Venkov ([14]) stated that (KerRes_{K}^{G})^{2} ⊂ (βu) (with β the Bockstein homomorphism), it turns out that, in general, (u, βu) is strictly smaller that KerRes_{K}^{G}; a counterexample is the case where G is any extraspecial p-group of order p^{3} and K is maximal elementary abelian in G (see [5,7]).

Quillen-Venkov’s result gives rise to the following questions:

(a) Given elements u_{1}, …, u_{m} of H^{1}(G) and K = ∩^{m}_{i=1}Keru_{i}, is there any relation between

KerRes_{K}^{G} and I_{K} = (βu_{1}, …, βu_{m}) (ignoring the inclusion I_{K} ⊂ KerRes_{K}^{G})?

(b) If (a) has an affirmative answer, is it possible to express an element of KerRes_{K}^{G} (or one of its power) via generators of I_{K}?

Consider now the Evens norm map N_{K→G} from H^{∗}(K) to H^{∗}(G) (under Benson’s construction in [1], N_{K→G}(η) can be defined for any η ∈ H^{∗}(K)). It was known that, given ξ ∈ H^{r}(G), the formula N_{K→G}Res_{K}^{G}(ξ) = ξ^{(G:K)} is generally false (see [4, Section 6, Remark 2]), while N_{K→G}Res_{K}^{G}(ξ) and ξ^{(G:K)} restrict to a same element of H^{∗}(K). With regards to N_{K→G}Res_{K}^{G}(ξ) − ξ^{(G:K)}, as an element of KerRes_{K}^{G}, the following theorem, stated in one of our earlier paper ([8]), gives a partial answer to the above questions.

Theorem 1.1. If K = Keru with u ∈ H^{1}(G), then

N_{K→G}Res_{K}^{G}(ξ) = \{ \begin{matrix}∑_{j}u^{j} · D_{j}(ξ) & , for p = 2 \\ ∑_{j}[(βu)^{j} · D_{2j}(ξ) + u(βu)^{j} · D_{2j+1}(ξ)] & , for p > 2 \end{matrix}

where, for p = 2 (resp. p > 2), D_{j} : H^{r}(G) → H^{r+j}(G) (resp. D_{j} : H^{r}(G) → H^{pr−j}(G)) are the Steenrod operations (composing with the Bocktein homomorphism if p is odd).

We shall give a generalization of the above theorem. By induction, it follows from the theorem that, for any subgroup K of G, N_{K→G}Res_{K}^{G}(ξ) can be expressed via the images of ξ under the Steenrod operations. In order to get such an expression, we need recall the Milnor basis of

Typeset by AMS-TEX

the Steenrod algebra A_{p}. First of all, A(p), as a vector space, has a basis consisting of the admissible monomials in Sq^{i} (resp. P^{i} and β) for p = 2 (resp. p > 2). Let A(p)^{∗} be the dual of A(p), and let ξ_{k} (resp. τ_{s} and ξ_{k}) be the duals of Sq^{2^{k−1}} … Sq^{2}Sq^{1} (resp. βP^{p^{s−1}} … P^{p}P^{1} and P^{p^{k−1}} … P^{p}P^{1}), with repect to that basis. In [6], Milnor showed that A(p)^{∗} is a commutative Hopf algebra, and, as an algebra

A(p)^{∗} = \{ \begin{matrix}P (ξ_{1}, ξ_{2}, …) & , for p = 2 \\ E(τ_{0}, τ_{1}, …) ⊗ P (ξ_{1}, ξ_{2}, …) & , for p > 2. \end{matrix}

It follows that A(p)^{∗} has the basis consisting of the monomials

\{ \begin{matrix}ξ^{R} & = ξ_{1}^{r_{1}} … ξ_{n}^{r_{n}} , for p = 2 \\ τ_{S}ξ^{R} & = τ_{s_{1}} … τ_{s_{k}}ξ_{1}^{r_{1}} … ξ_{n}^{r_{n}} , for p > 2 \end{matrix}

with S : s_{1} < ⋯ < s_{k}, R = (r_{1}, …, r_{n}, 0, …). For p = 2 (resp. p > 2), denote by St^{R} (resp. St^{S,R}) the dual of ξ^{R} (resp. τ_{S}ξ^{R}) with respect to this basis of A(p)^{∗}; the new basis of A(p) consisting of the (ξ^{R})^{∗} (resp. (τ_{S}ξ^{R})^{∗}) is called the Milnor basis. In [10,11], Mui showed that elements of this basis are nothing but the cohomology operations derived from modular invariants.

From now on, assume that K = ∩^{m}_{i=1}Keru_{i} is a subgroup of G with u_{i} ∈ H^{1}(G), 1 ≤ i ≤ m,

and let ξ be an element of H^{r}(G). For convenience, we also use the following notation. Given an element u ∈ H^{1}(G), set

ū = \{ \begin{matrix}u & , for p = 2 \\ βu & , for p > 2. \end{matrix}

Let

\begin{aligned}L_{m} & = L_{m}(ū_{1}, …, ū_{m}) \\ & = ∏^{m} ; i=1 ∏_{λ_{j}∈𝔽_{p}}(λ_{1}ū_{1} + ⋯ + λ_{i−1}ū_{i−1} + ū_{i}), \end{aligned}

and define the Dickson’s invariants Q_{m,s} = Q_{m,s}(ū_{1}, …, ū_{m}), 1 ≤ s ≤ m, recursively by (the variables will be omitted, if no confusion can arise):

\begin{aligned}Q_{m,0} & = L_{m}^{p−1}, \\ Q_{m,s} & = Q_{m−1,s}(L_{m}/L_{m−1})^{p−1} + Q_{m−1,s−1}^{p}, \end{aligned}

where Q_{s,s} = 1.

For p > 2 and for 0 ≤ s ≤ m − 1, set

\begin{aligned}M_{m,s} & = M_{m,s}(u_{1}, …, u_{m}; ū_{1}, …, ū_{m}) \\ & = | \begin{matrix}u_{1} & u_{2} & ⋯ & u_{m} \\ ū_{1} & ū_{2} & … & ū_{m} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ ū_{1}^{p^{s−1}} & ū_{2}^{p^{s−1}} & ⋯ & ū_{m}^{p^{s−1}} \\ ū_{1}^{p^{s+1}} & ū_{2}^{p^{s+1}} & ⋯ & ū_{m}^{p^{s+1}} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ ū_{1}^{p^{m−1}} & ū_{2}^{p^{m−1}} & ⋯ & ū_{m}^{p^{m−1}} \end{matrix} | , \end{aligned}

EVENS NORM AND RESTRICTION MAPS IN mod-p COHOMOLOGY OF p-GROUPS 3

and

\begin{aligned}L̃_{m} & = L_{m}^{h}, \\ M̃_{m,s} & = M_{m,s}L_{m}^{h−1}, \end{aligned}

with h = (p − 1)/2.

Generalizing Theorem 1.1, we prove

Theorem 1.2.

N_{K→G}Res_{K}^{G}(ξ) = \{ \begin{matrix}∑_{R=(r_{1},...,r_{m})}Q_{m,0}^{r−(r_{1}+···+r_{m})}Q_{m,1}^{r_{1}} ⋯ Q_{m,m−1}^{r_{m−1}} · St^{R}(ξ) & , for p = 2 \\ μ(r)^{m}_{∑_{R=(r_{1},...,r_{m})} ; 0≤s_{1}<···<s_{k}≤m−1}(−1)^{s(R,S)}M̃_{m,s_{1}} ⋯ M̃_{m,s_{k}} L̃_{m}^{r_{0}}Q_{m,1}^{r_{1}} \\ ⋯ Q_{m,m−1}^{r_{m−1}} · St^{S,R}(ξ) & , for p > 2 \end{matrix}

with μ(r) = (h!)^{r}(−1)^{hr(r−1)/2}, r_{0} = r − k − 2(r_{1} + ⋯ + r_{m}) and s(R, S) = k + s_{1} + ⋯ + s_{k} + r_{1} + 2r_{2} + ⋯ + mr_{m}.

We now derive some corollaries of the theorem. Note that, in general, the Evens norm map is not a ring homomorphism. However, we have

Corollary 1.3. N_{K→G}Res_{K}^{G} is a group homomorphism, and, up to a sign, a ring homomorphism.

The following is an answer to Questions (a) and (b).

Corollary 1.4. rad KerRes_{K}^{G} = rad I_{K}. Furthermore, if Res_{K}^{G}(ξ) = 0, then

ξ^{p^{m}} = \{ \begin{matrix}_{∑_{R=(r_{1},...,r_{m})} ; r≠r_{m}}Q_{m,0}^{r−(r_{1}+···+r_{m})}Q_{m,1}^{r_{1}} ⋯ Q_{m,m−1}^{r_{m−1}} · St^{R}(ξ) & , for p = 2 \\ _{∑_{R=(r_{1},...,r_{m})} ; r≠2r_{m} ; 0≤s_{1}<···<s_{k}≤m−1}(−1)^{s(R,S)+1}M̃_{m,s_{1}} ⋯ M̃_{m,s_{k}} L̃_{m}^{r_{0}}Q_{m,1}^{r_{1}} \\ ⋯ Q_{m,m−1}^{r_{m−1}} · St^{S,R}(ξ) & , for p > 2. \end{matrix}

Especially, if ξ is essential (i.e. ξ restricts trivially to all proper subgroups of G), it follows from the above corollary that ξ^{p} ∈ ∩_{0≠u∈H^{1}(G)}(u, βu); this fact was an effective tool in proving

that ξ^{p} = 0 if G is not elementary abelian (see [8]).

Denote by T_{G} the ideal of H^{∗}(G) generated by the images of transfers from proper subgroups of G. In the following corollary, we assume that u is a given element of H^{1}(G) and η = Res_{K}^{G}(ū).

Corollary 1.5. N_{K→G}(η) = ∑^{m} ; i=0ū^{p^{i}}Q_{m,i}. Furthermore, if G is extraspecial and K is maximal

elementary abelian in G, then there exists a power of N_{K→G}(η), hence of ū, which belongs to T_{G}.

The element ∑^{m} ; i=0ū^{p^{i}}Q_{m,i} is known to be the Mui’s invariant V_{m+1}(w_{1}, …, w_{m}, ū). The formula

relating N_{K→G}(η) and V_{m+1}(w_{1}, …, w_{m}, ū) in the above corollary is a direct consequence of Theorem 1.2 (in case m = 1). As we shall see in Section 2, when G is a direct product of K with an elementary abelian group, N_{K→G} can be expressed via the Steenrod power and the diagonal map, hence this formula was originally proved by Mui in [9, Chapter II, Lemma 5.3]. Later, in the context of the Evens norm map, it was also reproved by Okuyama-Sasaki ([12]).

The following notation will also be used. Let H(G) be the ideal of H^{∗}(G) consisting of elements of positive degree and let N_{K→G}^{+} be the image of H(K) via N_{K→G}. Given a subset S ⊂ H^{∗}(G) and elements ξ_{1}, …, ξ_{ℓ} of H^{∗}(G), denote by (ξ_{1}, …, ξ_{ℓ}, S) the ideal of H^{∗}(G) generated by ξ_{1}, …, ξ_{ℓ} and S. Set N_{G}^{+} = ∪_{M proper subgroup of G}N_{K→G}^{+} and let Φ(G) be the

Frattini subgroup of G. We have

Corollary 1.6.

H(G) = rad (ū_{1}, …, ū_{m}, N_{K→G}^{+})
= rad (βH^{1}(G), N_{Φ(G)→G}^{+})
= rad (N_{G}^{+}).

Let Z = Z(G) be the center of G. The last application is to get a new proof of the following result of Carlson.

Corollary 1.7 (Carlson [2]). rad KerRes_{Z}^{G} = rad T_{G}.

2. Evens norm and Steenrod power maps

Let E^{m} be an elementary abelian p-group of rank m and let W (E^{m}) be an E^{m}-free acyclic complex with augmentation ϵ : W (E^{m}) → 𝔽_{p}. Let x_{1}, …, x_{m} be a basis of H^{1}(E^{m}) and set

x̄_{i} = \{ \begin{matrix}x_{i} & , for p = 2 \\ βx_{i} & , for p > 2, \end{matrix}

1 ≤ i ≤ m. We have then

H^{∗}(E^{m}) = \{ \begin{matrix}𝔽_{2}[x_{1}, …, x_{m}] & , for p = 2 \\ Λ[x_{1}, …, x_{m}] ⊗ 𝔽_{p}[x̄_{1}, …, x̄_{m}] & , for p > 2. \end{matrix}

Consider E^{m} as the group of all translations on a vector space of dimension m over 𝔽_{p}. For every cochain complex C with diagonal C → C ⊗ C, set C^{V} = ⊗_{c∈C}C_{c}, C_{c} = C. We have the

chain map

H^{r}(C) ⟶^{P_{m}} H_{E^{m}}^{p^{m}r}(W (E^{m}) ⊗ K^{V}) ⟶^{d_{m}^{∗}} H^{∗}(E^{m}) ⊗ H^{∗}(C),

where P_{m} is the Steenrod power map, d_{m}^{∗} is induced by the diagonal C → C^{V} and the Kunneth formula. The following was proved in [15] (for m = 1) and [10,11] (for m ≥ 2).

Theorem 2.1.

(i) d_{m}^{∗}P_{m} = d_{m−s}^{∗}P_{m−s}d_{s}^{∗}P_{s}, 0 ≤ s ≤ m;

(ii) For ζ ∈ H^{r}(C), we have

d_{m}^{∗}P_{m}(ζ) = \{ \begin{matrix}∑_{R=(r_{1},...,r_{m})}Q_{m,0}^{r−(r_{1}+···+r_{m})}Q_{m,1}^{r_{1}} ⋯ Q_{m,m−1}^{r_{m−1}} ⊗ St^{R}(ζ) & , for p = 2 \\ μ(r)^{m}_{∑_{R=(r_{1},...,r_{m})} ; 0≤s_{1}<···<s_{k}≤m−1}(−1)^{s(R,S)}M̃_{m,s_{1}} ⋯ M̃_{m,s_{k}} L̃_{m}^{r_{0}}Q_{m,1}^{r_{1}} \\ ⋯ Q_{m,m−1}^{r_{m−1}} ⊗ St^{S,R}(ζ) & , for p > 2 \end{matrix}

with Q_{m,s} = Q_{m,s}(x̄_{1}, …, x̄_{m}) and M̃_{m,s} = M̃_{m,s}(x_{1}, …, x_{m}; x̄_{1}, …, x̄_{m}).

Consider the case where C is a cochain complex of which the cohomology is H^{∗}(G). We now relate d_{m}^{∗}P_{m} and the Evens norm map N_{G→G×E^{m}}. In so doing, we need recall some facts concerning the Evens norm map. Given a group Γ containing G as a subgroup, the Evens norm map was originally defined by Evens ([4]) as a map from ⊕_{n}H^{2n}(G) → ⊕_{n}H^{2n(Γ:G)}(Γ). Benson,

in [1], extends it to a map from ⊕_{n}H^{n}(G) → ⊕_{n}H^{n(Γ:G)}(Γ), providing an algebraic construction

of the Steenrod operations. Here are some properties of the Evens norm map (for details, the reader can refer to [1,4]).

Proposition 2.2.

(i) If K is a subgroup of G, then N_{K→Γ} = N_{G→Γ}N_{K→G}.

(ii) If H is a subgroup of Γ and Γ = ∐_{x∈D}GxH, then, for ζ ∈ H^{n}(G),

Res_{H}^{Γ}N_{G→Γ}(ζ) = ∏_{x∈D} N_{H∩^{x}G→H}Res_{H∩^{x}G}^{^{x}G}(^{x}ζ).

(iii) If G, G′ are respectively subgroups of Γ, Γ′ and f : Γ′ → Γ is a homomorphism such that f|_{G′} : G′ → G and f induces a bijection Γ′/G′ ≅ Γ/G of cosets spaces, then, for ζ ∈ H^{n}(G),

N_{G′→Γ′}(f|_{G′}^{∗}(ζ)) = f^{∗}(N_{G→Γ}(ζ)).

Especially, if N is a normal subgroup of Γ and N ⊂ G, then, for η ∈ H^{n}(G/N),

N_{G→Γ}Inf_{G}^{G/N}(η) = Inf_{Γ}^{Γ/N}N_{G/N→Γ/N}(η).

(iv) If ζ, ζ′ ∈ H^{n}(G), then

N_{G→Γ}(ζ + ζ′) = N_{G→Γ}(ζ) + N_{G→Γ}(ζ′)

+ a sum of transfers from proper subgroups of Γ containing the intersection of the conjugates of G.

(v) If ζ ∈ H^{r}(G), ζ′ ∈ H^{s}(G) and (Γ : G) = k, then

N_{G→Γ}(ζ · ζ′) = (−1)^{k(k−1)rs/2}N_{G→Γ}(ζ) · N_{G→Γ}(ζ′).

(vi) If N is a subgroup of G and ζ ∈ H^{n}(N), then N_{G→Γ}tr_{G}^{N}(ζ) is a sum of transfers from

subgroups L containing the intersection of the conjugates of N, and with L ∩ G ⊂ N. (vii) For ζ ∈ H^{n}(G),

N_{G→G×E^{1}}(ζ) = \{ \begin{matrix}x_{1}^{i} ⊗ Sq^{r−i}(ζ) & , for p = 2 \\ μ(r)_{∑_{e=0,1} ; 0≤2i≤r−e}(−1)^{e+i}x_{1}x̄_{1}^{(r−2i)h−e} ⊗ β^{e}P^{i}(ζ) & , for p > 2. \end{matrix}

It follows from Theorem 2.1 (ii) and Proposition 2.2 (vii) that N_{G→G×E^{1}} is nothing but d_{1}^{∗}P_{1}. Hence, from Theorem 2.1 (i), Proposition 2.2 (i), and by induction on m, we have

Corollary 2.3. d_{m}^{∗}P_{m} = N_{G→G×E^{m}}.

For later use, we also need the following

Corollary 2.4.

(i) If G is normal in Γ, then, for x ∈ Γ, ζ ∈ H^{∗}(G), we have N_{G→Γ}(^{x}ζ) = N_{G→Γ}(ζ).

(ii) If N is a normal elementary abelian subgroup of G and (G : C_{G}(N)) = p, then, for η ∈ KerRes_{N∩Z}^{N}, N_{N→G}(η) lies in T_{G}.

Proof. Part (i) follows from Proposition 2.2 (iii), by noting that H^{∗}(Γ) is invariant under the conjugation. We now prove (ii). Let n be the rank of N. By Proposition 2.2 (iv) and (v), we can suppose that η = ū with 0 ≠ u ∈ H^{1}(N). Pick an element a of G\C_{G}(N). Let Z_{0} = Z ⊂ Z_{1} ⊂ … be the upper central series of G and set N_{i} = N ∩ Z_{i}. A basis a_{1}, …, a_{n} of N can be chosen as follows: a_{1}, …, a_{n_{0}} is a basis of N_{0}, and, for i ≥ 0, a_{n_{i}+1}, …, a_{n_{i+1}} is a basis of a complement of N_{i} in N_{i+1}. It follows that, for n_{i} + 1 ≤ s ≠ t ≤ n_{i+1}, [a, a_{s}] and [a, a_{t}] are different elements of N_{i} and their images in G_{i−1} = G/Z_{i−1} belong to the center of G_{i−1}. Let v_{1}, …, v_{n} be the basis of H^{1}(N), dual to a_{1}, …, a_{n}. For n_{0} + 1 ≤ j ≤ n_{1}, there exists then w_{j} ∈ H^{1}(N) such that ^{a}w_{j} = w_{j} + v_{j}, hence ^{a}w̄_{j} = w̄_{j} + v̄_{j}; by (i) and by Proposition 2.2 (iv), this implies that N_{N→G}(v̄_{j}) ∈ T_{G}. For n_{i} ≤ j ≤ n_{i+1}, consider v_{j} as the image of an element of H^{1}(N/N_{i−1}), says t_{j}, via the inflation map Inf_{N}^{N/N_{i−1}}. Replacing G by G_{i−1}, the argument used before shows that N_{N/N_{i−1}→G_{i−1}}(t̄_{j}) ∈ T_{G_{i−1}}. Since the norm map and the transfer commute with the inflation, it follows that N_{N→G}(v̄_{j}) ∈ T_{G}. The corollary is proved.

3. Proofs of main results

Proof of Theorem 1.2. Fix a system of generators a_{1}, …, a_{k} of G satisfying u_{i}(a_{j}) = δ_{ij} with δ_{ij} the Kronecker symbol, 1 ≤ i ≤ m, 1 ≤ j ≤ k. Let c_{1}, …, c_{m} be a basis of E^{m} so that its dual is x_{1}, …, x_{m} and set Γ = G × E^{m}. Define a subgroup G′ of Γ by

G′ = ⟨a_{1}c_{1}^{−1}, …, a_{m}c_{m}^{−1}, a_{m+1}, …, a_{k}⟩.

It is clear that G′ is isomorphic to G and G ∩ G′ = K. Furthermore, the map f : Γ → Γ defined by

f(a_{j}) = \{ \begin{matrix}a_{j}c_{j}^{−1} & , for 1 ≤ j ≤ m \\ a_{j} & , for m + 1 ≤ j ≤ k \end{matrix}

is an automorphism of Γ satisfying f|_{G} : G ≅ G′ and f|_{K} = 1_{K} : K → K. By the Kunneth formula, we can identify H^{∗}(Γ) with H^{∗}(G′) ⊗ H^{∗}(E^{m}).

Let η be a Bar n-cochain of G′ representing (f^{−1})^{∗}ξ. Consider Res_{G}^{Γ}Inf_{Γ}^{G′}(η). Given elements g_{1}, …, g_{n} of G, we have

Res_{G}^{Γ}Inf_{Γ}^{G′}(η) = η(g_{1}, …, g_{n})
= η(f(g_{1}), …, f(g_{n}))
= f^{∗}η.

It follows that Res_{G}^{Γ}Inf_{Γ}^{G′}((f^{−1})^{∗}ξ) = ξ. Furthermore, as f|_{K} = 1_{K}, Res_{K}^{G′}((f^{−1})^{∗}ξ) = Res_{K}^{G}(ξ). Hence, from the commutative diagram

H^{∗}(G′) ⟶^{N_{G′→Γ}} H^{∗}(Γ) _{Res}_{K}^{G′} ↓  ↓^{Res}_{G}^{Γ} H^{∗}(K) ⟶_{N_{K→G}} H^{∗}(G) ,

and by Theorem 2.1 and Corollary 2.3, we have

N_{K→G}Res_{K}^{G}(ξ) = N_{K→G}Res_{K}^{G′}((f^{−1})^{∗}ξ)

= Res_{G}^{Γ}N_{G′→Γ}((f^{−1})^{∗}ξ)

= \{ \begin{matrix}Res_{G}^{Γ}( ∑_{R=(r_{1},...,r_{m})}Q_{m,0}^{r−(r_{1}+···+r_{m})}Q_{m,1}^{r_{1}} ⋯ Q_{m,m−1}^{r_{m−1}} ⊗ St^{R}((f^{−1})^{∗}ξ)) & , for p = 2 \\ Res_{G}^{Γ}(μ(r)^{m}_{∑_{R=(r_{1},...,r_{m})} ; 0≤s_{1}<···<s_{k}≤m−1}(−1)^{s(R,S)}M̃_{m,s_{1}} ⋯ M̃_{m,s_{k}} L̃_{m}^{r_{0}} \\ Q_{m,1}^{r_{1}} ⋯ Q_{m,m−1}^{r_{m−1}} ⊗ St^{S,R}((f^{−1})^{∗}ξ)) & , for p > 2, \end{matrix}

with Q_{m,s} = Q_{m,s}(ū_{1} + x̄_{1}, …, ū_{m} + x̄_{m}), M̃_{m,s} = M̃_{m,s}(u_{1} + x_{1}, …, u_{m} + x_{m}; ū_{1} + x̄_{1}, …, ū_{m} + x̄_{m}). The theorem is then proved, by noting that Res_{K}^{G′}((f^{−1})^{∗}ξ) = Res_{K}^{G}(ξ) and Res_{G}^{Γ}(u_{i}+x_{i}) = u_{i}, 1 ≤ i ≤ m.

Proofs of Corollaries 1.3 and 1.4. The proofs are straightforward from Theorem 1.2 and Proposition 2.2 (iv), (v), by noting that St^{R} and St^{S,R} are additive and St^{0,...,0,r}(ξ) = ξ^{2^{m}} for p = 2, and St^{∅,0...,0,r}(ξ) = ξ^{p^{m}} for p > 2.

Proof of Corollary 1.5. We need only prove the last assertion. Let w_{1}, …, w_{2n} be a basis of H^{1}(G/Z). Via the inflation, each w_{i} is also considered as an element of H^{1}(G). Denote by P the subalgebra of H^{∗}(G) generated by the w̄_{i}’s. Let v be a non-zero element of H^{1}(Z) and let A be the set of maximal elementary abelian subgroups of G. For every element A of A, A contains Z as a direct factor, so, via the inflation, v̄ can be considered as an element of H^{∗}(A). Set

ϕ = N_{K→G}(v̄ + η) − N_{K→G}(v̄).

From Proposition 2.2 (ii) and by a direct verification, it follows that Res_{A}^{G}(ϕ) = 0, for every A ∈ A. Hence ϕ is nilpotent. Furthermore, by Proposition 2.2 (iv), we have ϕ = N_{K→G}(η) mod T_{G}. Therefore, there exists a power of N_{K→G}(η) belongs to T_{G}.

Fix an element B of A and set ζ = ∑_{A∈A}N_{A→G}Res_{A}^{G}(ū). It follows from the first part of the

corollary that ζ ∈ P. Since every element of A is normal in G, by Proposition 2.2 (ii), we have

Res_{B}^{G}(ζ) = ∑_{A∈A} N_{A∩B→B}Res_{B∩A}^{G}(ū^{p^{n−k(A,B)}}),

with k(A, B) the codimension of A ∩ B in B. For every subspace C of B, let n_{C} be the number of elements A of A satisfying A ∩ B = C. By induction on the codimension of C in B, one can show that

n_{C} = \{ \begin{matrix}1 mod p & , for C = B \\ 0 mod p & , otherwise. \end{matrix}

Therefore Res_{B}^{G}(ζ) = Res_{B}^{G}(ū^{p^{n}}). Following [13, 16], Ker(P →^{Res} ∏_{A∈A}H^{∗}(A)) = {0}, so ζ = ū^{p^{n}}.

Since N_{A→G}Res_{A}^{G}(ū) belongs to rad T_{G}, for every A ∈ A, so does ū. The corollary is proved.

For convenience, given a variable X and a vector space V of dimension k over 𝔽_{p}, set

V (V; X) = V_{k+1}(t̄_{1}, …, t̄_{k}, X)

with t_{1}, …, t_{k} a basis of H^{1}(V) (we can easily show that V (V; X) does not depend on the choice of basis t_{1}, …, t_{k} of H^{1}(V)).

Proof of Corollary 1.6. We first prove that H(G) = (ū_{1}, …, ū_{m}, N_{K→G}^{+}). Set η = Res_{K}^{G}(ζ). Since η is invariant under the action of G/K, by Propsition 2.2 (ii),we have Res_{K}^{G}N_{K→G}(η) = η^{(G:K)} = η^{p^{m}}. Hence ψ = ζ^{p^{m}} − N_{K→G}(η) ∈ KerRes_{K}^{G}. By Corollary 1.4, there exists a power of ψ, says ψ^{p^{n}}, satisfying ψ^{p^{n}} ∈ (ū_{1}, …, ū_{m}). So ζ^{p^{n+m}} ∈ (ū_{1}, …, ū_{m}, N_{K→G}^{+}). Therefore H(G) ⊂ rad (ū_{1}, …, ū_{m}, N_{K→G}^{+}). The converse inclusion is obvious.

As Φ(G) = ∩_{0≠u∈H^{1}(G)}Keru, the second equality follows.

We now prove the third equality. Suppose that G is abelian. Let G′ be the maximal elementary abelian subgroup of G and let ξ ∈ H^{∗}(G). Consider the following cases:

(i) if G′ is a proper subgroup of G, then G′ is of form G′ = ∩^{m}_{i=1}Keru_{i}, with u_{i} ∈ H^{1}(G) satisfying

ū_{i}^{p} = 0, 1 ≤ i ≤ m. Hence

N_{G′→G}Res_{G′}^{G}(ξ^{p}) = ξ^{p(G:G′)}

which implies ξ ∈ rad N_{G}^{+}.

(ii) if G′ = G, then G is elementary abelian. Let n be the rank of G. Without loss of generality, we can assume that ξ = ū with u ∈ H^{1}(G). For every a ∈ G, let G_{a} be a linear complement in G of the subspace generated by a. Set

\begin{aligned}η & = ∑_{1≠a∈G}N_{⟨a⟩→G}Res_{⟨a⟩}^{G}(ū) \\ & = ∑_{1≠a∈G}V (G_{a}; ū). \end{aligned}

It follows from [3, Lemma 1.1] that η = ū^{p^{n−1}}. So ū ∈ rad N_{G}^{+}.

Finally, suppose that G is not abelian. Since the norm map commutes with the inflation, the argument used in the proof of Corollary 1.5 shows that there exists 0 ≠ u ∈ H^{1}(G) satisfying ū ∈ rad (N_{G}^{+}). By setting L = Keru, we have H(G) = rad (ū, N_{L→G}^{+}) ⊂ rad (N_{G}^{+}). The corollary is proved.

Proof of Corollary 1.7. It is clear that the corollary holds if G is abelian. So we can suppose that G is not abelian. Without loss of generality, we also suppose that Z(G) ⊂ Φ(G).

As H^{∗}(Z) is invariant under the action of G, a direct verification shows that Res_{Z}^{G}tr_{G}^{M} is the zero map, for every maximal subgroup M of G. So we need only prove that, for every 0 ≠ ξ ∈ H^{r}(G) ∩ KerRes_{Z′}^{G} with Z′ the maximal elementary abelian subgroup of Z, there exists a power of ξ belonging to T_{G}.

Assume that the corollary holds for every subgroup of G. Since the tranfer commutes with the inflation, Corollary 1.5 shows that there exists u ∈ H^{1}(G) satisfying ū ∈ rad T_{G}. Set K = Keru. K is then a maximal subgroup of G. Let W be the maximal elementary abelian subgroup of the center of K and set η = ξ^{p^{n}} − N_{W →G}Res_{W}^{G} (ξ), with p^{n} = (G : W). As Res_{K}^{G}(η) restricts trivially to the center of K, it follows from the inductive hypothesis that Res_{K}^{G}(η) ∈ rad T_{K}, hence, by Proposition 2.2 (vi), N_{K→G}Res_{K}^{G}(η) ∈ rad T_{G}. Furthermore, by Theorem 1.2, the fact that ū ∈ rad T_{G} and N_{K→G}Res_{K}^{G}(η) ∈ rad T_{G} implies η ∈ rad T_{G}. Since Res_{W}^{G} (ξ) satisfies the assumption of Corollary 2.4 (ii), it follows that N_{W →G}Res_{W}^{G} (ξ) ∈ T_{G}. Therefore ξ^{p^{n}}, hence ξ, belongs to rad T_{G}. The corollary is proved.

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Department of Mathematics, College of Sciences, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam

E-mail address: paminh@bdvn.vnd.net