Second cohomology group

Definition
Let $$G$$ be a group acting on an abelian group $$A$$, via an action $$\varphi:G \to \operatorname{Aut}(A)$$. Equivalently, $$A$$ is a module over the (possibly non-commutative) unital group ring $$\mathbb{Z}G$$ of $$G$$ over the ring of integers.

Definition in cohomology terms
The second cohomology group $$H^2_\varphi(G,A)$$ (also denoted $$H^2_\varphi(G;A)$$) is an abelian group defined in the following equivalent ways.

When $$\varphi$$ is understood from context, the subscript $${}_\varphi$$ may be omitted in the notation for the cohomology group, as well as the notation for the groups of 2-cocycles and 2-coboundaries.

All these definitions have natural analogues for the $$n^{th}$$ cohomology group $$H^n_\varphi(G,A)$$ for all $$n \ge 0$$. For more, see cohomology group.

Definition in terms of group extensions
There is an alternative definition of $$H^2_\varphi(G,A)$$ that is specific to 2 and has no easy analogue for other $$H^n_\varphi(G,A)$$. This is in terms of group extensions.

$$H^2_\varphi(G,A)$$ can also be identified with the set of congruence classes of group extensions with normal subgroup isomorphic to $$A$$ and quotient group isomorphic to $$G$$ where the induced action of the quotient is the specified action $$\varphi$$. By a group extension, we mean a group $$E$$ having $$A$$ as a normal subgroup and $$G$$ as a quotient group. Two extensions $$E_1$$ and $$E_2$$ are congruent if there is an isomorphism of $$E_1$$ to $$E_2$$ which is identity on $$A$$ and induces the identity map on $$G$$ as a quotient.

Particular cases
A very special case where a lot of additional things of interest happen is that where the action is trivial. See second cohomology group for trivial group action. In particular, in the case of a trivial action, the second cohomology group coincides with the second cohomology group of the classifying space of $$G$$ with coefficients in $$A$$.

Covariance in the second group
Suppose $$G$$ is a group and $$A_1,A_2$$ are abelian groups. Suppose $$\varphi_1:G \to \operatorname{Aut}(A_1)$$, $$\varphi_2:G \to \operatorname{Aut}(A_2)$$, and $$\alpha:A_1 \to A_2$$ are group homomorphisms such that $$\alpha \circ \varphi_1(g) = \varphi_2(g) \circ \alpha$$ for all $$g \in G$$.

In other words, $$\alpha$$ is a homomorphism from the $$G$$-module $$A_1$$ with action $$\varphi_1$$ to the $$G$$-module $$A_2$$ with action $$\varphi_2$$.

Then, we get an induced homomorphism between the second cohomology groups:

$$H^2(\alpha): H^2_{\varphi_1}(G,A_1) \to H^2_{\varphi_2}(G,A_2)$$

This association is functorial, i.e., it gives a (covariant) functor from the category of $$\mathbb{Z}G$$-modules (i.e., abelian groups with $$G$$ acting on them) to the category of abelian groups.

Contravariance in the first group
Suppose $$G_1,G_2$$ are groups and $$A$$ is an abelian group. Suppose $$\varphi_1:G_1 \to \operatorname{Aut}(A)$$, $$\varphi_2: G_2 \to \operatorname{Aut}(A)$$, and $$\alpha: G_1 \to G_2$$ are homomorphisms such that $$\varphi_2 \circ \alpha = \varphi_1$$, i.e., the $$G_1$$-action and $$G_2$$-action on $$A$$ are compatible. Then, we get an induced homomorphism between the second cohomology groups:

$$\operatorname{res}^{G_2}_{G_1}: H^2_{\varphi_2}(G_2,A) \to H^2_{\varphi_1}(G_1,A)$$

Note that the direction of this homomorphism is reverse to the direction of the original homomorphism. The association gives a contravariant functor. The functor in general is termed the restriction functor.

Automorphism group actions

 * Due to the covariance in the second argument, there is a natural action on $$H^2_\varphi(G,A)$$ of the group $$C_{\operatorname{Aut}(A)}(G) = C_{\operatorname{Aut}(A)}(\varphi(G))$$, i.e., the subgroup of the automorphism group of $$A$$ comprising those automorphisms that commute with the action of $$G$$.
 * Due to the contravariance in the first argument, there is a natural action on $$H^2_\varphi(G,A)$$ of the subgroup of $$\operatorname{Aut}(G)$$ that send every coset of the subgroup $$C_G(A)$$ to itself (or equivalently, induce the identity map on $$G/C_G(A)$$. Here, $$C_G(A)$$ is a normal subgroup of $$G$$ defined as the kernel of $$\varphi$$.

Examples
Here, we use the notation with $$G$$ a group acting on an abelian group $$A$$ via a group action $$\varphi$$.

Extreme examples

 * If $$G$$ is a trivial group, then the second cohomology group $$H^2_\varphi(G,A)$$ is also a trivial group.
 * If $$A$$ is a trivial group, then the second cohomology group $$H^2_\varphi(G,A)$$ is also a trivial group.