Skew of 2-cocycle for trivial group action of abelian group is alternating bihomomorphism
Contents
Statement
In terms of 2-cocycles
Let be a 2-cocycle for trivial group action (?) of an abelian group
on an abelian group
. In other words,
is a function such that:
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:
Related facts
Converse
Applications
- 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.
Proof
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:
and
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.