I'm trying to parse a statement I've found in The Cohomology of Groups by Leonard Evens.

For notation, let $G$ be a finite group, let $k$ be an algebraically closed field of characteristic $p>0$, and let $E$ be an elementary abelian $p$-subgroup of $G$. In section 9.2, Evens considers the object

$$\prod_{E\le G} H(E)$$

where the product is taken over all elementary abelian $p$-subgroups of $G$, and $H(E)$ is the commutative cohomology algebra of $E$ with coefficients in $k$, i.e.,


Evens then considers the subalgebra

$$I_G\subset \prod_{E\le G}H(E)$$

which consists of those "elements whose components are coherently related by restriction maps and conjugations."

I'm trying to understand the object $I_G$, and the following thought troubles me. Fix a particular elementary abelian $p$-subgroup $E$ such that $W_E:=N_G(E)/C_G(E)$ is nontrivial. Take nontrivial $g\in W_E$ so that via conjugation, $g$ induces a nontrivial map

$$g^*:H(gEg^{-1})=H(E)\to H(E)$$

If such an $E$ exists (with $W_E$ nontrivial), then wouldn't $I_G$ be empty? Suppose $I_G$ contains some element $\zeta$, and the component of $\zeta$ corresponding to $E$ is $\zeta_E$. Then if $g^*(\zeta_E)\ne \zeta_E$, we have that the components of $\zeta$ are not coherently related by conjugations. Then the only elements of $I_G$ are those $\zeta$ such that, for all $E\le G$, $\zeta_E$ is fixed by $g^*$ for all $g\in W_E$. Do such elements exist, and if so, is this the object Evens intends to define?


I was thinking about this lying in bed this morning, and I believe that $I_G$ does contain elements $\zeta$ such that for all $E$, $\zeta_E$ is fixed by $g^*$ for all $g\in W_E$.

Suppose $g\in W_E$ is not trivial. Then the following diagram commutes:

$$\require{AMScd}\begin{CD} E @>{c_g}>> E \\ @VVV @VVV \\ G@>{c_g}>> G \end{CD} $$

where $c_g$ is conjugation by $g$ and the vertical maps are inclusions. Since $G$ acts trivially on $H(G)$ via conjugation, we have the following induced diagram in cohomology:

$$\begin{CD} H(E) @<{g^*}<< H(E) \\ @AAA @AAA \\ H(G)@<{\operatorname{id}}<< H(G) \end{CD} $$

where the vertical maps are now restrictions. This shows that if we start with an element in $H(G)$, and let $\zeta_E$ be the restriction of this element to $H(E)$ for all $E$, then the tuple of cohomology classes thus defined is in fact coherently related via conjugations and restrictions.


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