Lemma 6

Let $c$ be a 2-cocycle of H with values in F. The image of $\widetilde{c}_s$ in the group $H^2(G,F)$ does not depend on the choice of the section $s$.

Let $s'$ be a section of the canonical surjection from G to $H\backslash G$. Let $h: H\backslash G\rightarrow H$ be the mapping characterized by the relation

for $x\in H\backslash G$. The 2-cocycle $\widetilde{c}_{s'}$ is then given by the relations

$s(x)^{-1}(-1''-1-1$ $=\prod_{x\in H\backslash G}^{g_1s(x\cdot g_1)^{-1}}c(h(x\cdot g_1)^{-1}, h(x\cdot g_1)s(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1})$ $s(x)^{-1}-1-1-1$ $\times \prod_{x\in H\backslash G}c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1))$ $=\prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1})$ $s(x)^{-1}-1-1-1$ $\times \prod_{x\in H\backslash G}c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1))$ $\times \prod_{x\in H\backslash G}^{g_1s(x)^{-1}}c(h(x)^{-1}, h(x)s(x)g_2s'(x\cdot g_2)^{-1})$ $s(x)^{-1}(-1''-1-1$

for all $g_1, g_2\in G$. The first equality comes from the cocycle relation (VIII, p. 295, formula (7)) applied to the elements

$h(x)^{-1},h(x)s(x)g_1s'(x\cdot g_1)^{-1}$, 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

$s(x)g_1s(x\cdot g_1)^{-1},h(x\cdot g_1)^{-1}$, and $h(x\cdot g_1)s(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1}$; and the last simply uses the fact that the mapping $x\mapsto x\cdot g_1$ is a permutation of $H\backslash G$.

The last two lines of the obtained expression correspond to a 2-coboundary. We find that $\widetilde{c}_{s'}$ has the same class in $H^2(G,F)$ as the cocycle whose value in $(g_1, g_2)\in G^2$ is given by the expression

$s(x)^{-1}-1-1-1$ $=\prod_{x\in H\backslash G}^{g_1s(x\cdot g_1)^{-1}}c(s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1}, h(x\cdot g_1g_2)^{-1})^{-1}$ $-1-1-1)$ $\times c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1))$ $=\prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1})$ $gs(x)^{-1}-1-1-1$

where the first equality follows from the cocycle relation applied to the elements

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

The last two lines of the obtained expression correspond to a 2-coboundary. We find that the class of $\widetilde{c}_{s'}$ coincides with that of $\widetilde{c}_s$.

We have constructed a homomorphism from the group $H^2(H,F)$ to the group $H^2(G,F)$, which we call the corestriction homomorphism from H to G and denote by Cor$^G_H$.