Second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4

Description of the group
We consider here the second cohomology group for trivial group action of the direct product of Z4 and Z2 on the cyclic group:Z4, i.e.,

$$\! H^2(G,A)$$

where $$G \cong \mathbb{Z}_4 \times \mathbb{Z}_2$$ and $$A \cong \mathbb{Z}_4$$.

The cohomology group is isomorphic to direct product of Z4 and V4.

Computation of cohomology group
The cohomology group can be computed as an abstract group using the group cohomology of direct product of Z4 and Z2.

The general formula for $$H^2(G;A)$$ for this choice of $$G$$ is:

$$A/4A \oplus A/2A \oplus \operatorname{Ann}_A(2)$$

In this case, with $$A = \mathbb{Z}/2\mathbb{Z}$$, this becomes:

$$\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$$

This is direct product of Z4 and V4.

Elements
We list here the elements, grouped by similarity under the action of the automorphism groups on both sides.

In all cases, order of extension group is product of order of normal subgroup and quotient group, so the order is $$4 \times 8 = 32$$. Also, since these are central extensions (because the action is trivial), the nilpotency class of the extension group is at least 1 (the nilpotency class of the quotient) and at most 2.

The minimum size of generating set is at least 2 (the minimum size of generating set for the quotient) and at most 3 (the sum of the minimum size of generating set for the normal subgroup and the quotient). See minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group and minimum size of generating set of quotient group is at most minimum size of generating set of group.

None of the extensions are stem extensions, because the base cyclic group:Z4 is not a quotient of the Schur multiplier of the acting group, which is cyclic group:Z2.

General background
We know from the general theory that there is a natural short exact sequence:

$$0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0$$

where the image of $$\operatorname{Ext}^1$$ is $$H^2_{sym}(G;A)$$, i.e., the group of cohomology classes represented by symmetric 2-cocycles (and corresponding to the abelian group extensions). We also know, again from the general theory, that the short exact sequence above splits, i.e., $$H^2_{sym}(G;A)$$ has a complement inside $$H^2$$. However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

In this case
In terms of the general background, one way of putting this is that the skew map:

$$H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A)$$

has a section (i.e., a reverse map):

$$\operatorname{Hom}(\bigwedge^2G,A) \to H^2(G;A)$$

whose image is $$H^2_{CP}(G;A)$$ of cohomology classes represented by cyclicity-preserving 2-cocycles (see cyclicity-preserving subgroup of second cohomology group for trivial group action). Thus, the natural short exact sequence splits, and we get an internal direct sum decomposition:

$$H^2(G;A) = H^2_{sym}(G;A) + H^2_{CP}(G;A)$$

A pictorial description of this is as follows. Here, each column is a coset of $$H^2_{CP}(G,A)$$ and each row is a coset of $$H^2_{sym}(G,A)$$. The top left entry is the identity element, hence the top row corresponds to abelian group extensions and the left column corresponds to cyclicity-preserving 2-cocycles.

The group $$\operatorname{Aut}(G) \times \operatorname{Aut}(A)$$ permutes columns 2-5 transitively and columns 7-8 transitively.

Generalized Baer Lie rings
The direct sum decomposition (discussed in the preceding section):

$$H^2(G;A) = H^2_{sym}(G;A) + H^2_{CP}(G;A)$$

gives rise to some examples of the cocycle halving generalization of Baer correspondence. For any group extension arising as an element of $$H^2(G;A)$$, the additive group of its Lie ring arises as the group extension corresponding to the projection onto $$H^2_{sym}(G;A)$$, and the Lie bracket coincides with the group commutator.

In the description below, the additive group of the Lie ring of a given group is the unique abelian group in the column corresponding to that group.

We thus get the following correspondences: