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

Definition

Let $G$ be a group acting (on the left) on an abelian group $A$ via a homomorphism of groups $\varphi :G\to \operatorname {Aut} (A)$ where $\operatorname {Aut} (A)$ is the automorphism group of $A$.

Explicit definition

A 2-cocycle for the action is a function $f:G\times G\to A$ satisfying:

$\!\varphi (g_{1})(f(g_{2},g_{3}))+f(g_{1},g_{2}g_{3})=f(g_{1}g_{2},g_{3})+f(g_{1},g_{2})$

If we suppress $\varphi$ and use $\cdot$ for the action, we can rewrite this as:

$\!g_{1}\cdot f(g_{2},g_{3})+f(g_{1},g_{2}g_{3})=f(g_{1}g_{2},g_{3})+f(g_{1},g_{2})$ or equivalently:

$\!g_{1}\cdot f(g_{2},g_{3})-f(g_{1}g_{2},g_{3})+f(g_{1},g_{2}g_{3})-f(g_{1},g_{2})=0$

Note that a function $f:G\times G\to A$ (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 $n=2$. 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

The set of 2-cocycles for the action of $G$ on $A$ forms a group under pointwise addition. This group is denoted $Z_{\varphi }^{2}(G,A)$ where $\varphi$ is the action. If the action is understood from the context, it can simply be denoted as $Z^{2}(G,A)$.

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 2-cocycles $f:G\times G\to A$ can be identified with the group of $\mathbb {Z} G$-module maps from $K$ to $A$.

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
$G$ acts trivially on $A$ (i.e., every element of $G$ fixes every element of $A$)	$\!f(g_{2},g_{3})+f(g_{1},g_{2}g_{3})=f(g_{1}g_{2},g_{3})+f(g_{1},g_{2})$	2-cocycle for trivial group action
$G$ is abelian with additive notation for its group operation, and it acts trivially on $A$	$\!f(g_{2},g_{3})+f(g_{1},g_{2}+g_{3})=f(g_{1}+g_{2},g_{3})+f(g_{1},g_{2})$

Examples

Extreme examples

• If $A$ 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 $G$ to the zero element of $A$.
• If $G$ is the trivial group, the group of 2-cocycles is isomorphic to the group $A$, with each 2-cocycle being identified with its image value in $A$.

Other examples

Acting group $G$	Group $A$ acted upon	Action	Group of 2-cocycles
trivial group	any group $A$	trivial action	isomorphic to $A$, where a 2-cocycle is identified with its image point
any group $G$	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	$f(g,h):=g\cdot \alpha (h)-\alpha (gh)+\alpha (g)$ for some function $\alpha :G\to A$	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	$f$ is a 2-cocycle and is a symmetric function of both its inputs
normalized 2-cocycle for a group action	$f$ is a 2-cocycle and if either of the inputs to $f$ is the identity element of $f$, the output of $f$ is zero

Importance

Extension involving an abelian normal subgroup

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

$\!s(gh)=f(g,h)s(g)s(h)$

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

Such an $f$ is a 2-cocycle.

Note that for a particular choice of $E$, all the 2-cocycles obtained by different choices of $S$ 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 $E$ with abelian normal subgroup $A$ and quotient $G$, an element of the second cohomology group.

Further information: second cohomology group