2-cocycle for trivial group action
Explicitly, it is a function satisfying the following condition:
The set of 2-cocycles for trivial group action form a group, denoted . 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
|Property||Meaning||Proof of implication||Proof of strictness||Capture of difference||Intermediate notions|
|2-coboundary for trivial group action||There exists such that||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 such that for all|||FULL LIST, MORE INFO|
|skew-symmetric 2-cocycle for trivial group action||A 2-cocycle such that for all|||FULL LIST, MORE INFO|
|normalized 2-cocycle for trivial group action (also called identity-preserving 2-cocycle)||A 2-cocycle such that if either of the inputs to is the identity element, the output of is the identity element|||FULL LIST, MORE INFO|
|cyclicity-preserving 2-cocycle for trivial group action||A 2-cocycle such that whenever is cyclic||IIP 2-cocycle for trivial group action|FULL LIST, MORE INFO|
|bihomomorphism with both input groups identical (equal to ) and the output group abelian (equal to )||A function such that, if either input for is fixed, is a homomorphism from the other input to||bihomomorphism to abelian group implies 2-cocycle|||FULL LIST, MORE INFO|
Extensions involving a central subgroup
Let be a group with a central subgroup isomorphic to (and explicitly identified with) , and a quotient isomorphic to (and explicitly identified with) , 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 be a system of coset representatives for in with being the representation map. Then, define such that
In other words, measures the extent to which the collection of coset representatives fails to be closed under multiplication.
Such an is a 2-cocycle for trivial group action of on .
Note that for a particular choice of , all the 2-cocycles obtained by different choices of 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 with abelian normal subgroup and quotient , an element of the second cohomology group.
Further information: second cohomology group for trivial group action