1-cocycle for a group action

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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

• 1-cocycle for a Lie ring action

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 $A$.