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

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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 (?).