Homology group for trivial group action commutes with direct product in second coordinate

From Groupprops
Jump to: navigation, search

Statement

For two groups

Suppose G is a group and A_1 and A_2 are (possibly isomorphic, possibly non-isomorphic) abelian groups. Let q be a nonnegative integer. Denote by H_q(G;A_1), H_q(G;A_2), and H_q(G;A_1 \times A_2) the homology groups for trivial group action of G on A_1, A_2, and the external direct product A_1 \times A_2 respectively. Then, there is a canonical isomorphism:

H_q(G;A_1) \times H_q(G;A_2) \cong H_q(G;A_1 \times A_2)