Group of cyclicity-preserving 2-cocycles for trivial group action is naturally identified with group of homomorphisms from a particular abelian group
Suppose is a group. Then, there exists an abelian group such that, for any abelian group , the group of cyclicity-preserving 2-cocycles for the trivial group action can be identified with the group of homomorphisms under pointwise addition.
Examples of 2-groups
When is a 2-group, is also a 2-group.
Examples of other p-groups
We consider the case .
|Group||Order||Prime-base logarithm of order||Group that acts as source of homomorphisms||Order||Prime-base logarithm of order|
|elementary abelian group:E9||9||2||elementary abelian group:E27||27||3|
|direct product of Z9 and Z3||27||3||elementary abelian group:E243||243||5|