Second cohomology group for trivial group action

Definition

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 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

Facts

• Formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization

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:

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

Specific examples

Acting group $G$	Group $A$ acted upon	Second cohomology group	Groups obtained as extensions	More information
cyclic group:Z2	cyclic group:Z2	cyclic group:Z2	Klein four-group and cyclic group:Z4	second cohomology group for trivial group action of Z2 on Z2
cyclic group:Z2	cyclic group:Z4	cyclic group:Z2	direct product of Z4 and Z2 and cyclic group:Z8	second cohomology group for trivial group action of Z2 on Z4
cyclic group:Z2	Klein four-group	Klein four-group	elementary abelian group:E8, direct product of Z4 and Z2 (occurs in three ways)	second cohomology group for trivial group action of Z2 on V4
cyclic group:Z4	cyclic group:Z2	cyclic group:Z2	direct product of Z4 and Z2 and cyclic group:Z8	second cohomology group for trivial group action of Z4 on Z2
Klein four-group	cyclic group:Z2	elementary abelian group:E8	elementary abelian group:E8, direct product of Z4 and Z2 (3 times), dihedral group:D8 (3 times), quaternion group	second cohomology group for trivial group action of V4 on Z2
cyclic group:Z2	cyclic group:Z8	cyclic group:Z2	direct product of Z8 and Z2 and cyclic group:Z16
cyclic group:Z4	cyclic group:Z4	cyclic group:Z4	direct product of Z4 and Z4, cyclic group:Z16 (occurs in two ways), direct product of Z8 and Z2
Klein four-group	cyclic group:Z4	elementary abelian group:E8	direct product of Z4 and V4, direct product of Z8 and Z2, central product of D8 and Z4, M16	second cohomology group for trivial group action of V4 on Z4
Klein four-group	Klein four-group	elementary abelian group:E64	elementary abelian group:E16, direct product of Z4 and Z4, direct product of Z4 and V4, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, direct product of D8 and Z2, direct product of Q8 and Z2	second cohomology group for trivial group action of V4 on V4
Klein four-group	cyclic group:Z8	elementary abelian group:E8	direct product of Z8 and V4, central product of D8 and Z8, direct product of Z16 and Z2, M32	second cohomology group for trivial group action of V4 on Z8