# 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 $G$ is a 2-group that is an Elementary abelian group (?) and $A$ is an abelian group with the property that every element of order $2$ in $A$ is a square. Then, we have: $\! H^2(G,A) = H^2_{sym}(G,A) + H^2_{CP}(G,A)$

In words, the Second cohomology group for trivial group action (?) of $G$ on $A$ 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 $G$ is an $2$-group that is an elementary abelian group and $A$ is an abelian group with the property that every element of order $2$ in $A$ is a square. Then, every central extension with base $A$ and quotient $G$ has its own generalized Baer lie ring via the Cocycle skew reversal generalization of Baer correspondence (?).