# 1-coboundary for a group action

*This term is defined, and makes sense, in the context of a group action on an Abelian group. In particular, it thus makes sense for a linear representation of a group*

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This article gives a basic definition in the following area: group cohomology

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## Definition

Let be a group acting on an Abelian group , i.e., there exists a homomorphism of groups where is the automorphism group of .

A **1-coboundary**, also called a **principal crossed homomorphism**, for this group action is a function such that there exists a such that for all :

If we denote the action by , this can be rewritten as:

The 1-coboundaries form an Abelian group under pointwise addition of functions.

## Importance

Suppose is a group having as a normal subgroup with as the quotient group. Then, 1-coboundaries in correspond to inner automorphisms by elements of which are in the stability group of the ascending series .

In fact, the for the inner automorphism and the coboundary is the same.

The 1-coboundary group is thus a quotient of the group itself, by the subgroup of comprising -invariant elements. Two particular cases:

- The action of on is faithful: In this case the 1-coboundary group is isomorphic to
- The action of on is trivial: In this case the 1-coboundary group is trivial

`Further information: 1-cocycle for a group action`