Proposition 8

Let H be a subgroup of G of finite index. The endomorphism Cor$^G_H\circ$ Res$^G_H$ of the group $H^2(G,F)$ coincides with multiplication by the index (G : H).

Let $\alpha$ be an element of $H^2(G,F)$, and let $c$ be an element of $Z^2(G,F)$ representing $\alpha$. The element Cor$^G_H\circ$ Res$^G_H(\alpha )$ is the class of the cocycle whose value in $(g_1, g_2)\in G^2$ is given by the expression

$=\prod_{x\in H\backslash G}c(g_1, g_2s(x\cdot g_1g_2)^{-1})c(g_1, s(x\cdot g_1)^{-1})^{-1}$

The first equality comes from the cocycle relation (VIII, p. 295, formula (7)) applied to the elements

$s(x)^{-1},s(x)g_1s(x\cdot g_1)^{-1}$, and $s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1}$;

the second is obtained by applying the cocycle relation to the elements

$g_1,s(x\cdot g_1)^{-1}$, and $s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1}$.

By removing a coboundary, we obtain that Cor$^G_H\circ$ Res$^G_H(\alpha )$ is the class of the cocycle whose value in $(g_1, g_2)\in G^2$ is given by the expression

which proves the result.