# Second cohomology group for trivial group action

## Contents

## Definition

### Context for definition

Let be a group and 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 on . This group is denoted .

Note that 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 , is defined as the quotient where is the group of 2-cocycles for the trivial group action and is the group of 2-coboundaries for the trivial group action.

### Definition in terms of group extensions

can also be identified with the set of congruence classes of central extensions of by , i.e., group extensions where the normal subgroup is a central subgroup and the quotient group is .

### Definition in terms of cohomology of classifying space

Suppose is the classifying space of . The **second cohomology group for trivial group action** is defined as the second cohomology group 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 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

## 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: