Second cohomology group for trivial group action of V4 on Z2

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:Z2, i.e.,

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

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

The cohomology group is isomorphic to elementary abelian group:E8.

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 also $$\mathbb{Z}/2\mathbb{Z}$$. 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
Each element of the second cohomology group corresponds to a group extension with base normal subgroup cyclic group:Z2 in the center and the quotient group isomorphic to Klein four-group. Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each extension group is $$2 \times 4 = 8$$.

Further, the minimum size of generating set of the extension group is at least equal to 2 (the minimum size of generating set of the quotient Klein four-group) and at most equal to 3 (the sum of the minimum size of generating set for the normal subgroup and quotient group).

Explicit description and relation with power-commutator presentation
Consider an extension group $$E$$ with central subgroup isomorphic to $$A$$ (cyclic group:Z2) and quotient group $$G$$ isomorphic to Klein four-group. Denote by $$\overline{a_1}, \overline{a_2}$$ a basis for $$G$$ (i.e., two unequal non-identity elements of $$G$$) and by $$a_1,a_2$$ elements of $$E$$ that map to $$\overline{a_1},\overline{a_2}$$ respectively. Denote by $$a_3$$ a non-identity element of the central subgroup.

Then, $$E$$ is generated by the elements $$a_1,a_2,a_3$$. Further, we can get a power-commutator presentation for $$E$$ using these generators. Specifically, we know that $$[a_1,a_3] = e, [a_2,a_3] = e, a_3^2 = e$$. We also know that the elements $$a_1^2, a_2^2, [a_1,a_2]$$ are each either equal to $$e$$ or to $$a_3$$.

In order to specify the cohomology class of the extension, it is necessary and sufficient to specify, for each of $$a_1^2, a_2^2, [a_1,a_2]$$, whether it equals $$a_3$$ or $$e$$. In terms of the notation for the power-commutator presentation, this is equivalent to saying that $$\beta(1,2) = 0$$ and each of $$\beta(1,3), \beta(2,3), \beta(1,2,3)$$ can be either 0 or 1, and are viewed as elements of cyclic group:Z2. Here:


 * $$\beta(1,3)$$ is the power of $$a_3$$ that $$a_1^2$$ equals. It is 0 if $$a_1^2 = e$$ (i.e., is the identity element) and 1 if $$a_1^2 = a_3$$.
 * $$\beta(2,3)$$ is the power of $$a_3$$ that $$a_2^2$$ equals. It is 0 if $$a_2^2 = e$$ (i.e., is the identity element) and 1 if $$a_2^2 = a_3$$.
 * $$\beta(1,2,3)$$ is the power of $$a_3$$ that $$[a_1,a_2]$$ equals. It is 0 if $$[a_1,a_2] = e$$ (i.e., is the identity element) and 1 if $$[a_1,a_2] = a_3$$.

The total number of possibilities is $$2^3$$. Further, the mapping from $$H^2(G,A)$$ that sends a cohomology class to the tuple $$(\beta(1,3),\beta(2,3),\beta(1,2,3))$$ is an isomorphism of additive groups. This means that to add two cohomology classes, we can add the corresponding tuples.

Here is a more detailed explanation for why the mapping described here is well defined up to cohomology and gives an isomorphism:


 * Explanation for why the values $$\beta(1,3), \beta(2,3), \beta(1,2,3)$$ depend only on the cohomology class: What we need to show is that the values $$\beta(1,3),\beta(2,3),\beta(1,2,3)$$ are invariant under replacing $$a_1$$ or $$a_2$$ by the other coset representatives of the central subgroup in $$E$$, i.e., the values remain the same if we replace $$a_1$$ by $$a_1a_3$$ and/or we replace $$a_2$$ by $$a_2a_3$$. We check these below:
 * Check for $$\beta(1,3)$$: We need to show that $$(a_1a_3)^2 = a_1^2$$. Here's why: $$(a_1a_3)^2 = a_1a_3a_1a_3 = a_1^2a_3^2 = a_1^2$$ (we use that $$a_3$$ is central and has order two).
 * Check for $$\beta(2,3)$$: We need to show that $$(a_2a_3)^2 = a_2^2$$. Here's why: $$(a_2a_3)^2 = a_2a_3a_2a_3 = a_2^2a_3^2 = a_2^2$$ (we use that $$a_3$$ is central and has order two).
 * Check for $$\beta(1,2,3)$$: This follows from the fact that the commutator of two elements depends only on their cosets mod the center.
 * Explanation for why each of the value mappings $$\beta(1,3), \beta(2,3), \beta(1,2,3)$$ is a homomorphism to cyclic group:Z2: For this we need to return to the original description of cocycle addition.
 * Explanation for injectivity of the mapping to the tuple $$(\beta(1,3), \beta(2,3), \beta(1,2,3))$$: Since the mapping is a homomorphism, it suffices to show that the kernel of the mapping is trivial. This is easy to see: if $$\beta(1,3) = \beta(2,3) = \beta(1,2,3) = 0$$ then the cohomology class of the extension must be trivial.
 * Explanation for surjectivity of the mapping to the tuple $$(\beta(1,3), \beta(2,3), \beta(1,2,3))$$:

We provide below the full list of elements. Note that $$\beta(1,2) = 0$$ in all cases:

Generalizations

 * Second cohomology group for trivial group action of V4 on finite cyclic 2-group
 * Second cohomology group for trivial group action of elementary abelian group of prime-square order on group of prime order

Summary of action
By pre-composition, the automorphism group of the Klein four-group (which is isomorphic to symmetric group:S3) acts on the second cohomology group. Under this action, there are four orbits, corresponding to the four types of group extensions given above. The elementary abelian group:E8 and quaternion group are fixed points. Specifically, the 3 cohomology classes that give direct product of Z4 and Z2 are in one orbit, while the 3 cohomology classes that give dihedral group:D8 are in another orbit.

Description of group action in terms of explicit descriptions of elements
The discussion here relies on the explicit description of cohomology classes in terms of the invariants $$\beta(1,3), \beta(2,3), \beta(1,2,3)$$.

The automorphism group of the Klein four-group acts as symmetric group:S3 on the elements $$\overline{a_1}, \overline{a_2}, \overline{a_1}\overline{a_2}$$. This shuffling leads to a corresponding change in the values of $$\beta(1,3)$$ and $$\beta(2,3)$$. The invariant $$\beta(1,2,3)$$, which determines whether the extension group is abelian, is invariant under this transformation. However, the value of $$\beta(1,2,3)$$ affects the way in which the automorphism affects $$\beta(1,3)$$ and $$\beta(2,3)$$.

Trivial outer action
We consider the classification of congruence classes of extensions where the base normal subgroup has center isomorphic to cyclic group:Z2 and the quotient group is isomorphic to the Klein four-group, and the quotient has trivial outer action on the base. In this case, the set of congruence classes of extensions is classified by $$H^2(V_4;\mathbb{Z}_2)$$, with the identity element corresponding to the external direct product.

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. 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
For this choice of $$G$$ and $$A$$, the subgroup $$H^2_{CP}(G;A)$$ of cyclicity-preserving cohomology classes is trivial, hence the sum $$H^2_{sym}(G;A) + H^2_{CP}(G;A)$$ is $$H^2_{sym}(G;A)$$ and is not the whole group $$H^2(G;A)$$. Thus, $$H^2_{CP}(G;A)$$ is not the desired complement.

However, there is a complement that we call $$J$$ to $$H^2_{sym}(G;A)$$ in $$H^2(G;A)$$. This comprises the trivial extension and the extension that gives the quaternion group. Further, this complement is invariant under the action of $$\operatorname{Aut}(G) \times \operatorname{Aut}(A)$$. As an internal direct sum:

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

as an internal direct sum. A pictorial description of this would be as follows. Here, each column is a coset of $$J$$ 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 $$J$$.

The group $$\operatorname{Aut}(G) \times \operatorname{Aut}(A)$$ acts as symmetric group:S3 on the three columns other than the left column. Each of the rows is preserved.

Size information
We first give some quantitative size information if we use non-normalized cocycles and coboundaries:

In particular, what this means is that for every cohomology class, there are 4 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 2 different choices of normalized 2-cocycles that represent that cohomology class.

Finding a group of cocycle representatives
Consider the short exact sequence for cocycles and coboundaries:

$$0 \to B^2(G;A) \to Z^2(G;A) \to H^2(G;A) \to 0$$

and the corresponding one for normalized cocycles and coboundaries:

$$0 \to B^2_n(G;A) \to Z^2_n(G;A) \to H^2(G;A) \to 0$$

Since these are short exact sequences of vector spaces, they must split. Further, a splitting of the latter also gives a splitting of the former.

There is no automorphism-invariant choice of splitting (i.e., no choice of splitting that is invariant under the $$\operatorname{Aut}(G) \times \operatorname{Aut}(A)$$-action). However, we can define splittings once we have established a basis for $$G$$.

Use the same notation as in the section on explicit descriptions and power-commutator presentations. Thus, $$G$$ has basis $$\overline{a_1}, \overline{a_2}$$ and $$A$$ has elements $$\{ e, a_3 \}$$ with $$e$$ the identity element and $$a_3$$ the non-identity element. Then, for each cohomology class, pick the unique representative normalized 2-cocycle $$f$$ such that:

$$\! f(\overline{a_1},\overline{a_2}) = e$$

or, if we denote the group additively with $$a_3 = 1, e = 0$$:

$$\! f(\overline{a_1},\overline{a_2}) = 0$$

What this basically says, in terms of coset representative choices, is that we are making sure that the coset representative for $$\overline{a_1}\overline{a_2}$$ equals $$a_1a_2$$. In other words, we are pinning down the ambiguity in picking one of the non-identity coset representatives. Our choice of normalized already eliminates the ambiguity with picking the identity coset representative. The computations above show that this picks a unique coset representative.

Further, by the pointwise nature of cocycle addition, these uniquely chosen coset representatives form a subgroup.

Linear choices of cocycle and algebra group structures
The upshot is that all the cohomology classes can be represented by bilinear choices of 2-cocycle, and there are two choices for each cohomology class. A bilinear choice of 2-cocycle means that the group has the structure of an adjoint group for the corresponding radical ring. In this case, because everything is happening over field:F2, we get algebra group structures over the field. To summarize: there are two algebra group structures for the extension group corresponding to each cohomology class. In all the algebra group structures, the corresponding algebra has nilpotency index at most three, i.e., all products of length three in each of the algebras is zero.

Note that the extension groups may have other algebra groups unrelated to their realization in terms of these cohomology classes.

Homomorphisms on $$A$$
The unique injective homomorphism from $$A = \mathbb{Z}_2$$ to $$\mathbb{Z}_4$$ induces a homomorphism:

$$H^2(G;A) \to H^2(G;\mathbb{Z}_4)$$

The group on the right is also isomorphic to elementary abelian group:E8 (see second cohomology group for trivial group action of V4 on Z4). However, the induced map above is not an isomorphism. Rather, it has kernel $$H^2_{sym}(G,A)$$ (comprising all the abelian extensions) and its image is the cyclicity-preserving subgroup of second cohomology group for trivial group action $$H^2_{CP}(G,\mathbb{Z}_4)$$ that comprises the trivial group extension (direct product of Z4 and V4) and the extension central product of Q8 and Z4.

In terms of extensions, the map is interpreted as follows: it involves taking the central product of a given extension with cyclic group:Z4, identifying the base cyclic group:Z2 in the original extension with the $$\mathbb{Z}_2$$ in $$\mathbb{Z}_4$$.

The map is given in the table below.

The unique surjective homomorphism from $$\mathbb{Z}_4$$ to $$A = \mathbb{Z}_2$$ induces a homomorphism:

$$H^2(G,\mathbb{Z}_4) \to H^2(G,A)$$

The kernel of this map is the cyclicity-preserving subgroup of second cohomology group for trivial group action $$H^2_{CP}(G,\mathbb{Z}_4)$$ that comprises the trivial group extension (direct product of Z4 and V4) and the extension central product of Q8 and Z4. The image is the group $$H^2_{Sym}(G,\mathbb{Z}_2)$$. Thus, the roles of kernel and image are reversed from the previous map.

Homomorphism on $$G$$
For each injective map from $$\mathbb{Z}_2$$ to $$G$$, we get a corresponding restriction homomorphism:

$$H^2(G,A) \to H^2(\mathbb{Z}_2,A)$$

The kernel of each of these homomorphisms is a Klein four-group comprising one copy of elementary abelian group:E8, one copy of direct product of Z4 and Z2, and two copies of dihedral group:D8. Each of the Klein four-groups is different and $$\operatorname{Aut}(G) \times \operatorname{Aut}(A)$$ permutes them transitively. The only element not in any of the kernels is the element corresponding to the quaternion group extension.

For each surjective map from $$G$$ to $$\mathbb{Z}_2$$, we get a corresponding inflation homomorphism:

$$H^2(\mathbb{Z}_2,A) \to H^2(G,A)$$

For each of these, the image of the homomorphism is a cyclic group:Z2 comprising the trivial extension elementary abelian group:E8 and one copy of the symmetric nontrivial extension direct product of Z4 and Z2. Each of these image subgroups is different, and their union is the whole of $$H^2_{sym}(G,A)$$. The automorphism group $$\operatorname{Aut}(G) \times \operatorname{Aut}(A)$$ permutes them transitively.

Construction of the cohomology group
The cohomology group can be constucted using the GAP functions ElementaryAbelianGroup, TwoCohomology, TrivialGModule, GF.

gap> G := ElementaryAbelianGroup(4);; gap> A := TrivialGModule(G,GF(2));; gap> T := TwoCohomology(G,A); rec( group := , module := rec( field := GF(2), isMTXModule := true, dimension := 1, generators := [ ,  ] ), collector := rec( relators := [ [ 0 ], [ [ 2, 1 ], 0 ] ], orders := [ 2, 2 ], wstack := [ [ 1, 1 ], [ 2, 1 ] ], estack := [ ], pstack := [ 3, 3 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ], list := [ 0, 0 ], module := [ ,  ], mone := , mzero := , avoid := [ ], unavoidable := [ 1, 2, 3 ] ), cohom :=  -> ( GF(2)^3 )>, presentation := rec( group := , relators := [ f1^2, f1^-1*f2*f1*f2^-1, f2^2 ] ) )

Construction of extensions
The extensions can be constructed using the additional command Extensions.

gap> G := ElementaryAbelianGroup(4);; gap> A := TrivialGModule(G,GF(2));; gap> L := Extensions(G,A);; gap> List(L,IdGroup); [ [ 8, 5 ], [ 8, 2 ], [ 8, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 2 ], [ 8, 3 ], [ 8, 4 ] ] gap> FrequencySort(last); [ [ [ 8, 2 ], 3 ], [ [ 8, 3 ], 3 ], [ [ 8, 4 ], 1 ], [ [ 8, 5 ], 1 ] ]

Construction of automorphism group actions
This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives.

gap> G := ElementaryAbelianGroup(4);; gap> A := TrivialGModule(G,GF(2));; gap> A1 := AutomorphismGroup(G);; gap> A2 := GL(1,2);; gap> D := DirectProduct(A1,A2);; gap> P := CompatiblePairs(G,A,D);; gap> M := ExtensionRepresentatives(G,A,P);; gap> List(M,IdGroup); [ [ 8, 5 ], [ 8, 2 ], [ 8, 3 ], [ 8, 4 ] ]