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 $G$ be a group acting on an Abelian group $A$, i.e., there exists a homomorphism of groups $\varphi:G \to \operatorname{Aut}(A)$ where $\operatorname{Aut}(A)$ is the automorphism group of $A$.

A 1-coboundary, also called a principal crossed homomorphism, for this group action is a function $f:G \to A$ such that there exists a $a \in A$ such that for all $g \in G$:

$f(g) = \varphi(g)(a) - a$

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

$f(g) = g \cdot a - a$

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

Importance

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

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

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

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

Further information: 1-cocycle for a group action