# 2-cocycle for trivial group action

## Definition

Suppose $G$ is a group and $A$ is an abelian group. A 2-cocycle for trivial group action for $G$ on $A$ is a 2-cocycle for the trivial group action of $G$ on $A$.

Explicitly, it is a function $f:G \times G \to A$ satisfying the following condition:

$\! f(g,hk) + f(h,k) = f(gh,k) + f(g,h) \ \forall \ g,h,k, \in G$

The set of 2-cocycles for trivial group action form a group, denoted $Z^2(G,A)$. Note that the same notation is used for the group of 2-cocycles for a nontrivial group action as well, though in the latter case, a subscript for the action may be used or the specific action is made clear from the context.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness Capture of difference Intermediate notions
2-coboundary for trivial group action There exists $\alpha:G \to A$ such that $f(g,h) := \alpha(g) + \alpha(h) - \alpha(gh)$ The quotient of the group of 2-cocycles by the group of 2-coboundaries is the second cohomology group for the trivial action |FULL LIST, MORE INFO
symmetric 2-cocycle for trivial group action A 2-cocycle $f$ such that $\! f(g,h) = f(h,g)$ for all $g,h \in G$ |FULL LIST, MORE INFO
skew-symmetric 2-cocycle for trivial group action A 2-cocycle $f$ such that $\! f(g,h) + f(h,g) = 0$ for all $g,h \in G$ |FULL LIST, MORE INFO
normalized 2-cocycle for trivial group action (also called identity-preserving 2-cocycle) A 2-cocycle $f$ such that if either of the inputs to $f$ is the identity element, the output of $f$ is the identity element |FULL LIST, MORE INFO
cyclicity-preserving 2-cocycle for trivial group action A 2-cocycle $f$ such that $f(g,h) = 0$ whenever $\langle g,h \rangle$ is cyclic IIP 2-cocycle for trivial group action|FULL LIST, MORE INFO
bihomomorphism with both input groups identical (equal to $G$) and the output group abelian (equal to $A$) A function $f:G \times G \to A$ such that, if either input for $f$ is fixed, $f$ is a homomorphism from the other input to $A$ bihomomorphism to abelian group implies 2-cocycle |FULL LIST, MORE INFO

## Importance

### Extensions involving a central subgroup

Let $E$ be a group with a central 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 on the subgroup (in the sense of action by conjugation, see quotient group acts on abelian normal subgroup). 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 for trivial group action of $G$ on $A$.

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 for trivial group action