# 2-cocycle for a group action

*This term is defined, and makes sense, in the context of a group action on an Abelian group. In particular, it thus makes sense for a linear representation of a group*

## Contents

## Definition

Let be a group acting (on the left) on an abelian group via a homomorphism of groups where is the automorphism group of .

### Explicit definition

A 2-cocycle for the action is a function satisfying:

If we suppress and use for the action, we can rewrite this as:

or equivalently:

Note that a function (without any conditions) is sometimes termed a 2-cochain for the group action.

### Definition as part of the general definition of cocycle

A 2-cocycle for a group action is a special case of a cocycle for a group action, namely . This, in turn, is the notion of cocycle corresponding to the Hom complex from the bar resolution of to as -modules.

### Group structure

The set of 2-cocycles for the action of on forms a group under pointwise addition. This group is denoted where is the action. If the action is understood from the context, it can simply be denoted as .

### As a group of homomorphisms

For any group , we can construct a -module such that for any abelian group , the group of 2-cocycles can be identified with the group of -module maps from to .

`Further information: group of 2-cocycles is naturally identified with group of homomorphisms from a particular module`

## Particular cases and variations

Case or variation | Condition for 2-coycle | Further information |
---|---|---|

acts trivially on (i.e., every element of fixes every element of ) | 2-cocycle for trivial group action | |

is abelian with additive notation for its group operation, and it acts trivially on |

## Examples

### Extreme examples

- If is the trivial group, the group of 2-cocycles is the trivial group, with the only 2-cocycle being the map that sends every pair of elements of to the zero element of .
- If is the trivial group, the group of 2-cocycles is isomorphic to the group , with each 2-cocycle being identified with its image value in .

### Other examples

Acting group | Group acted upon | Action | Group of 2-cocycles |
---|---|---|---|

trivial group | any group | trivial action | isomorphic to , where a 2-cocycle is identified with its image point |

any group | trivial group | trivial action | trivial group -- the unique 2-cocycle that sends everything to the zero element |

cyclic group:Z2 | cyclic group:Z2 | trivial action | ? |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Capture of difference | Intermediate notions |
---|---|---|---|---|---|

2-coboundary for a group action | for some function | 2-coboundary implies 2-cocycle | 2-cocycle not implies 2-coboundary | The quotient of the group of 2-cocycles by the group of 2-coboundaries is termed the second cohomology group | |FULL LIST, MORE INFO |

symmetric 2-cocycle for a group action | is a 2-cocycle and is a symmetric function of both its inputs | ||||

normalized 2-cocycle for a group action | is a 2-cocycle and if either of the inputs to is the identity element of , the output of is zero |

## Importance

### Extension involving an abelian normal subgroup

Let be a group with an abelian normal subgroup isomorphic to (and explicitly identified with) , and a quotient isomorphic to (and explicitly identified with) , such that the induced action of the quotient on (in the sense of action by conjugation, see quotient group acts on abelian normal subgroup) is as described. Let be a system of coset representatives for in with being the representation map. Then, define such that

In other words, measures the extent to which the collection of coset representatives fails to be closed under multiplication.

Such an is a 2-cocycle.

Note that for a particular choice of , all the 2-cocycles obtained by different choices of will form a single coset of the coboundary group, that is, any two such cocycles will differ by a coboundary. Thus, in particular, we can intrinsically associate, to every extension with abelian normal subgroup and quotient , an element of the second cohomology group.

`Further information: second cohomology group`