Normalized 2-cocycle for trivial group action

Definition

Suppose $G$ is a group and $A$ is an abelian group. A normalized 2-cocycle for trivial group action is a function $f:G\times G\to A$ that satisfies both these conditions:

• $f$ is a 2-cocycle for trivial group action: $f(g,hk)+f(h,k)=f(gh,k)+f(g,h)\mathrm{\forall }g,h,k,\in G$
• $f$ evaluates to the identity element of $A$ (written as 0 since $A$ is abelian) if either of its inputs is the identity element of $G$: $f(g,1)=f(1,g)=0\mathrm{\forall }g\in G$

The set of normalized 2-cocycles for trivial group action form a subgroup (under pointwise addition) of the group of 2-cocycles for trivial group action. Although this is a proper subgroup whenever $A$ is nontrivial, it still maps surjectively to the second cohomology group for trivial group action, as explained in the Importance section.

Importance

Normalizing a 2-cocycle

Given a 2-cocycle for trivial group action $f:G\times G\to A$, we can obtain a normalized 2-cocycle $f_1:G\times G\to A$ defined as follows:

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Relation with treatment as central extensions

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 $A$. The different possible 2-cocycles for a given central extension form a single coset of the subgroup of 2-coboundaries, and hence a single element of the second cohomology group for trivial group action $H^2(G,A)$.

If we further restrict $s$ so that $s(1)$ is the identity element of $A$, i.e., the coset representative for the base subgroup is its identity element, then the 2-cocycle we obtain this way is a normalized 2-cocycle.

Since it's always possible to choose $s$ so that $s(1)$ is the identity element of $A$, every central extension can be represented by a normalized 2-cocycle. In other words, every element of the second cohomology group for trivial group action $H^2(G,A)$ can be represented by a normalized 2-cocycle. Another way to frame this is that $Z^2(G,A)$ (the group of all 2-cocycles for trivial group action) is generated by $B^2(G,A)$ (the subgroup comprising 2-coboundaries) and the subgroup comprising normalized 2-cocycles.