# Orthogonal direct sum of cocycles across acting groups

From Groupprops

## Definition

Suppose and are natural numbers. Consider a bunch of groups , all equipped with actions on an abelian group . Suppose, for each , that we have a -cocycle for the action of on .

Then, consider . If all the -actions on commute with each other, then we get an induced -action on . We can define a -cocycle for this action in terms of s, as the **orthogonal direct sum of cocycles** as follows:

First, view as an internal direct product, so each is identified with a subgroup of . Now, define:

where the summation on the right happens in .

This does indeed give a cocycle.