1-cocycle for a group action

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This article gives a basic definition in the following area: group cohomology
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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

Definition

A 1-cocycle is defined in the context of a group G, an abelian group A, and an action of G on A by automorphisms, i.e., a homomorphism of groups \varphi:G \to \operatorname{Aut}(A) where \operatorname{Aut}(A) is the automorphism group of A.

Explicit definition

A 1-cocycle of G in A, also called a crossed homomorphism from G to A, is a function f:G \to A satisfying:

\! f(gh) = f(g) + \varphi(g)(f(h))

Here the group operation in G is expressed multiplicatively, and the group operation in A is expressed additively.

If we suppress \varphi and simply denote the action by \cdot, the condition is:

\! f(gh) = f(g) + g \cdot f(h)

Definition as part of the general definition of cocycle

A 1-cocycle for a group action is a special case of a cocycle for a group action in the case n = 1. This, in turn, is the notion of cocycle corresponding to the Hom complex from the bar resolution of G to A as \mathbb{Z}G-modules.

Group structure

For a group G acting on an abelian group A, the set of 2-cocycles for the action of G on A forms a group under pointwise addition.

As a group of homomorphisms

For any group G, we can construct a \mathbb{Z}G-module K such that for any abelian group A, the group of 1-cocycles f:G \times G \to A can be identified with the group of \mathbb{Z}G-module maps from K to A.

Particular cases and variations

Case or variation Condition for a function to be a 1-coycle Further information
Action is trivial \! f(gh) = f(g) + f(h), so f is a homomorphism of groups from G to A. In fact, it descends to a homomorphism from the abelianization G/[G,G] to A
G is an abelian group, hence is written additively \! f(g + h) = f(g) + g \cdot f(h)

Related notions

Importance

As stability automorphisms for an extension

Suppose we are given a group G acting on an abelian group A, and a group E which is the semidirect product of A by G with respect to this action. Now consider the group of those automorphisms of E which fix A pointwise and which are identity on G (viewed as a quotient).

The claim is that this group is isomorphic to the group of 1-cocycles. The isomorphism is as follows. The automorphism takes every coset of A to itself, and acts as identity on A. Hence it must simply translate each coset of A by an element of A. Consider the function f:G \to A by the map f(g) = g^{-1}\sigma(g), in other words, this describes the amount by which each coset gets translated. The fact that \sigma is an automorphism forces f to be a 1-cocycle.

Interestingly, it is also true that the 1-coboundaries correspond to those stability automorphisms that arise as inner automorphisms by elements of A. Refer 1-coboundary for a group action.

The quotient of these groups (the first cohomology group) thus measures the group of stability automorphisms of 1 \triangleleft A \triangleleft E upto inner automorphisms of A.