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