Second cohomology group for trivial group action

Context for definition
Let $$G$$ be a group and $$A$$ be an abelian group.

In terms of more general definition
The second cohomology group for trivial group action is defined as the defining ingredient::second cohomology group for the trivial group action of $$G$$ on $$A$$. This group is denoted $$H^2(G,A)$$.

Note that $$H^2(G,A)$$ is also used for the more general notion of second cohomology group with an accompanying action. The notation is interpreted in terms of the trivial group action only if that is explicitly stated or is otherwise clear from context.

Definition in terms of explicit 2-cocycles and 2-coboundaries
The second cohomology group, denoted $$H^2(G,A)$$, is defined as the quotient $$Z^2(G,A)/B^2(G,A)$$ where $$Z^2(G,A)$$ is the group of 2-cocycles for the trivial group action and $$B^2(G,A)$$ is the group of 2-coboundaries for the trivial group action.

Definition in terms of group extensions
$$H^2(G,A)$$ can also be identified with the set of congruence classes of central extensions of $$A$$ by $$G$$, i.e., group extensions where the normal subgroup $$A$$ is a central subgroup and the quotient group is $$G$$.

Definition in terms of cohomology of classifying space
Suppose $$BG$$ is the classifying space of $$G$$. The second cohomology group for trivial group action $$H^2(G,A)$$ is defined as the second cohomology group $$H^2(BG,A)$$ where the latter is in the sense of the cohomology of a topological space (for instance, singular or cellular cohomology).

Group actions on the second cohomology group

 * Automorphism group of base group acts on second cohomology group for trivial group action: Note that there is a corresponding statement for a nontrivial group action, but in that more general case, we can only make the subgroup $$C_{\operatorname{Aut}(A)}(G)$$ act.
 * Automorphism group of acting group acts on second cohomology group for trivial group action

Subgroups of interest
Some subgroups of interest are:


 * IIP subgroup of second cohomology group for trivial group action
 * cyclicity-preserving subgroup of second cohomology group for trivial group action

Examples

 * Second cohomology group for trivial group action of finite cyclic group on finite cyclic group
 * Second cohomology group for trivial group action commutes with direct product in second coordinate: There is a natural isomorphism:

$$\! H^2(G,A_1 \times A_2) \cong H^2(G,A_1) \times H^2(G,A_2)$$