1-coboundary for a group action

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