Cohomology group for trivial group action

Definition
Let $$G$$ be a group and $$A$$ be an abelian group.

The cohomology groups for trivial group action $$\! H^n(G,A)$$, also denoted $$\! H^n(G;A)$$ ($$n = 0,1,2,3,\dots$$) are abelian groups defined in the following equivalent ways.

Definition in terms of classifying space
$$\! H^n(G,A)$$ can be defined as the cohomology group $$H^n(BG,A)$$, where $$BG$$ is the classifying space of $$G$$ and the cohomology group is understood to be in the topological sense (singular cohomology or cellular cohomology, or any of the equivalent cohomology theories satisfying the axioms).

Definitions as cohomology group for an action taken as the trivial action
The cohomology groups for trivial group action $$H^n(G,A)$$ are defined as the defining ingredient::cohomology groups $$H^n_\varphi(G,A)$$ where $$\varphi:G \to \operatorname{Aut}(A)$$ is the trivial map. In other words, we treat $$A$$ as a $$G$$-module with trivial action of $$G$$ on $$A$$ (i.e., every element of $$G$$ fixes every element of $$A$$. We thus also treat $$A$$ as a trivial $$\mathbb{Z}G$$-module, where $$\mathbb{Z}G$$ is a group ring of $$G$$ over the ring of integers $$\mathbb{Z}$$.

The definitions below are basically adaptations of the general definitions of cohomology group to the case where the action is trivial.

Equivalence of definitions
To show the equivalence of definitions between the topological and algebraic definitions, we can proceed as follows:


 * Prove the equivalence of definitions of homology groups with coefficients in the integers.
 * Note that the dual universal coefficients theorem holds in both cases, and can be used to get natural isomorphisms.