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: $\!\varphi(g_1)(f(g_2,g_3)) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_1,g_2)$

If we suppress $\varphi$ and use $\cdot$ for the action, we can rewrite this as: $\!g_1 \cdot f(g_2,g_3) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_1,g_2)$ or equivalently: $\! g_1 \cdot f(g_2,g_3) - f(g_1g_2,g_3) + f(g_1,g_2g_3) - f(g_1,g_2) = 0$

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 $Z^2_\varphi(G,A)$ 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 $A$.

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 $A$) $\! f(g_2,g_3) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_1,g_2)$ 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