Second cohomology group for trivial group action of V4 on Z4

Description of the group
We consider here the second cohomology group for trivial group action of the specific information about::Klein four-group on specific information about::cyclic group:Z4, i.e.,

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

where $$G \cong V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$$ and $$A \cong \mathbb{Z}_4$$.

The cohomology group is isomorphic to elementary abelian group:E8. As a vector space over field:F2, it is a three-dimensional vector space.

Computation in terms of group cohomology
The cohomology group can be computed as an abstract group using the group cohomology of Klein four-group, which in turn can be computed using the Kunneth formula for group cohomology combined with the group cohomology of cyclic group:Z2.

We explain here the part of the computation based on the group cohomology of Klein four-group. As per that page, we have:

$$H^2(G;A) = (\operatorname{Ann}_A(2)) \oplus (A/2A)^2$$

Here, $$A/2A$$ is the quotient of $$A$$ by $$2A = \{ 2x \mid x \in A \}$$ and $$\operatorname{Ann}_A(2) = \{ x \in A \mid 2x = 0 \}$$.

In our case, $$A = \mathbb{Z}/2\mathbb{Z}$$, so we get that both $$A/2A$$ and $$\operatorname{Ann}_A(2)$$ are both isomorphic to $$\mathbb{Z}/2\mathbb{Z}$$. Plugging in, we get:

$$H^2(G;A) = \mathbb{Z}/2\mathbb{Z} \oplus (\mathbb{Z}/2\mathbb{Z})^2 = (\mathbb{Z}/2\mathbb{Z})^3$$

which is the elementary abelian group of order eight.

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

Explicit description and relation with power-commutator presentation
Consider an extension group $$E$$ with central subgroup isomorphic to $$A$$ (cyclic group:Z4) and quotient group $$G$$ isomorphic to Klein four-group. Denote by $$\overline{a_1}, \overline{a_2}$$ elements of $$G$$ that form a basis for $$G$$ (i.e., they are unequal non-identity elements). Denote by $$a_3$$ a generator for $$A$$ and by $$a_4$$ the element $$a_3^2$$.

Let $$a_1,a_2$$ be elements of $$E$$ that map to $$\overline{a_1}, \overline{a_2}$$ respectively.

Then, $$a_1,a_2,a_3,a_4$$ generate $$E$$. Moreover, we can construct a power-commutator presentation for $$E$$ using these generators. Specifically, we know that $$[a_1,a_3] = [a_1,a_4] = e, a_3^2 = a_4, a_4^2 = e$$ and $$[a_1,a_2],a_1^2, a_2^2$$ are all in the set $$\{ e,a_3,a_3a_4,a_4 \}$$. Moreover, we also have that $$[a_1,a_2]^2 = [a_1^2,a_2]$$ and this tells us that in fact $$[a_1,a_2] \in \{ e, a_4 \}$$. The upshot is that we know that $$\beta(1,2) = \beta(1,2,3) = \beta(2,3,4) = \beta(1,3,4) = 0$$ and we need to specify the values of $$\beta(1,3), \beta(1,4), \beta(2,3), \beta(2,4), \beta(1,2,4)$$. This is a total of five numbers.

Some further whittling down is necessary. Note that by replacing $$a_1$$ by $$a_1a_3$$ if necessary, we can make sure that $$\beta(1,4) = 0$$. Similarly, by replacing $$a_2$$ by $$a_2a_3$$ is necessary, we can make sure that $$\beta(2,4) = 0$$. Thus, we have three values $$\beta(1,3), \beta(2,3), \beta(1,2,4)$$ that need to be specified. The mapping that sends a cohomology class to this triple of values, each viewed as an element of cyclic group:Z2, is an isomorphism.

The generic presentation we have is:

$$\langle a_1,a_2,a_3,a_4 \mid a_1^2 = a_3^{\beta(1,3)}, a_2^2 = a_3^{\beta(1,3)}, a_3^2 = a_4, a_4^2 = e, [a_1,a_2] = a_4^{\beta(1,2,4)}, [a_1,a_3] = [a_2,a_3] = [a_1,a_4] = [a_2,a_4] = [a_3,a_4] = e \rangle$$

Based on the choices of $$\beta(1,3), \beta(2,3), \beta(1,2,4)$$, we get an actual presentation and an actual extension. The table below lists the possibilities:

Action of automorphism group of acting group
By pre-composition, the automorphism group of the Klein four-group acts on the second cohomology group. Under this action, there are four orbits, corresponding to the four group extensions given above. Specifically, the 3 cohomology classes that give direct product of Z8 and Z2 are in one orbit, while the 3 cohomology classes that give M16 are in another orbit.

Action of automorphism group of base group
The automorphism group of the base group has no effect on the cohomology classes. This is because this automorphism, the inverse map, pulls back trivially to the acting group, which has exponent two.

Description of group actions in terms of explicit descriptions of elements
The automorphism group of the Klein four-group can be thought of as $$2 \times 2$$ matrices, and these naturally act on a two-dimensional vector space with basis $$\beta(1,3), \beta(2,3)$$. This is the desired action. $$\beta(1,2,4)$$ is fixed by the automorphism group.

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 automorphism action acts as symmetric group:S3 on the three columns other than the left column, but preserves rows.

Size information
In particular, what this means is that for every cohomology class, there are 64 different choices of 2-cocycles that represent that cohomology class.

We give the corresponding information if we use normalized cocycles and coboundaries:

In particular, what this means is that for every cohomology class, there are 16 different choices of normalized 2-cocycles that represent that cohomology class.

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.

Thus, we have two correspondences emerging: