Skew of 2-cocycle for trivial group action of abelian group is alternating bihomomorphism
In terms of 2-cocycles
Then, the function:
is an alternating bihomomorphism from to .
Interpretation in terms of second cohomology group
Since any 2-coboundary for the trivial group action is a symmetric 2-cocycle for trivial group action, its skew is zero. Thus, the skew of a 2-cocycle depends only on its cohomology class. The above statement can thus be described by saying that gives a homomorphism from the second cohomology group to the group of alternating bihomomorphisms:
- Class two implies commutator map is endomorphism: In a group of nilpotency class two, the commutator map is an endomorphism in either variable. This is a corollary of the fact stated here, if we interpret as the center, as the inner automorphism group, and is a 2-cocycle representing the extension. The skew of is the commutator map.
Given: An abelian group and an abelian group . A 2-cocycle .
To prove: The function is an alternating bihomomorphism.
Proof: Clearly, for all , so it suffices to show that:
Since is a 2-cocycle from to , we have:
Since this is true for all , the corresponding statement is true with cyclically permuted:
Interchanging and in the original expression, we get:
We now do (1) + (2) - (3) to obtain:
Since the variables are universally quantified, this proves the right linearity. Because of the left right symmetry, it also proves right linearity.