# Second cohomology group for trivial group action is internal direct sum of symmetric and cyclicity-preserving 2-cocycle subgroups if acting group is elementary abelian 2-group and every element of order two in the base group is a square

## Statement

### In terms of internal direct sums

Suppose is a 2-group that is an Elementary abelian group (?) and is an abelian group with the property that every element of order in is a square. Then, we have:

In words, the Second cohomology group for trivial group action (?) of on is the internal direct product (internal direct sum) of the subgroup generated by the images of symmetric 2-cocycles (which corresponds to the extensions that are abelian) and the Cyclicity-preserving subgroup of second cohomology group for trivial group action (?) (which is the image in cohomology of the group of cyclicity-preserving 2-cocycles).

### In terms of existence of generalized Baer cyclicity-preserving Lie ring

Suppose is an -group that is an elementary abelian group and is an abelian group with the property that every element of order in is a square. Then, every central extension with base and quotient has its own generalized Baer lie ring via the Cocycle skew reversal generalization of Baer correspondence (?).