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 under pointwise addition, 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.

Facts

• 2-cocycle for trivial group action is constant on axes: For a 2-cocycle for trivial group action $f:G\times G\to A$, $f(g,1)=f(1,g)=f(1,1)$ for all $g\in G$, where $1$ denotes the identity element of $G$. This fact is crucial for setting the stage for the concept of normalized 2-cocycle for trivial group action.
• 2-cocycle for trivial group action is symmetric between element and inverse: For a 2-cocycle for trivial group action $f:G\times G\to A$, $f(g,g^{-1})=f(g^{-1},g)$ for all $g\in G$. This fact is crucial for setting the stage for the concept of IIP 2-cocycle for trivial group action.

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			The quotient of the group of 2-cocycles by the subgroup of normalized 2-cocycles is isomorphic to $A$, corresponding to the value of the 2-cocycle at $(1,1)$	|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, Normalized 2-cocycle for trivial group action|FULL LIST, MORE INFO
IIP 2-cocycle for trivial group action	A 2-cocycle $f$ such that $f(g,g^{-1})=0$ for all $g$				Normalized 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 $A$.

It turns out that the reverse is also true: given any 2-cocycle for trivial group action, we can construct a central extension that realizes it. The idea described above is constructive and can be used to define the central extension.

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 2-coboundary subgroup, that is, any two such cocycles will differ by a 2-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. The association is bijective, so the second cohomology group corresponds to the central extensions.

Further information: second cohomology group for trivial group action