Homology group for trivial group action commutes with direct product in second coordinate
From Groupprops
Statement
For two groups
Suppose is a group and
and
are (possibly isomorphic, possibly non-isomorphic) abelian groups. Let
be a nonnegative integer. Denote by
,
, and
the homology groups for trivial group action of
on
, and the external direct product
respectively. Then, there is a canonical isomorphism: