# Second cohomology group for trivial group action commutes with direct product in second coordinate

## Statement

Suppose $G$ is a group and $A_1$ and $A_2$ are abelian groups. Let $H^2(G,A_1)$, $H^2(G,A_2)$, and $H^2(G,A_1 \times A_2)$ denote the Second cohomology group for trivial group action (?) for $G$ on $A_1$, $A_2$, and the External direct product (?) $A_1 \times A_2$ respectively. Then, we have the following natural isomorphism:

$\! H^2(G,A_1 \times A_2) \cong H^2(G,A_1) \times H^2(G,A_2)$

## Particular cases

$G$ $A_1$ $A_2$ $A_1 \times A_2$ $H^2(G,A_1)$ $H^2(G,A_2)$ $H^2(G,A_1 \times A_2)$
cyclic group:Z2 cyclic group:Z2 cyclic group:Z2 Klein four-group cyclic group:Z2 (cohomology information) cyclic group:Z2 (cohomology information) Klein four-group (cohomology information)
Klein four-group cyclic group:Z2 cyclic group:Z2 Klein four-group elementary abelian group:E8 (cohomology information) elementary abelian group:E8 (cohomology information) elementary abelian group:E64 (cohomology information)
cyclic group:Z4 cyclic group:Z2 cyclic group:Z2 Klein four-group cyclic group:Z2 (cohomology information) cyclic group:Z2 (cohomology information) Klein four-group (cohomology information)