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

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This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group direct product of Z4 and Z4 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and direct product of Z4 and Z4 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
Get more specific information about direct product of Z4 and Z4 |Get more specific information about cyclic group:Z2

Description of the group

This group is the second cohomology group for trivial group action of the homocyclic group direct product of Z4 and Z4 on cyclic group:Z2. In other words, it is the group:

H^2(G,A)

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

This group is isomorphic to elementary abelian group:E8.

Elements

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 32) Is the base normal subgroup characteristic?
trivial 1 direct product of Z4 and Z4 and Z2 21 No
symmetric and nontrivial 3 direct product of Z8 and Z4 3 Yes
non-symmetric 3 semidirect product of Z8 and Z4 of M-type 4 Yes
non-symmetric 1 SmallGroup(32,2) 2 Yes

Group actions

Under the action of the automorphism group of the acting group

The automorphism group of the acting group permutes transitively all the elements of a given cohomology class type. In particular, the trivial cohomology class and the class giving SmallGroup(32,2) are fixed points, whereas the classes giving direct product of Z8 and Z4 form one orbit and the cohomology classes giving semidirect product of Z8 and Z4 of M-type give another orbit.

Subgroups of interest

Subgroup Value Corresponding group extensions for subgroup GAP IDs Group extension groupings for each coset GAP IDs
subgroup generated by images of symmetric 2-cocycles (corresponds to abelian group extensions) Klein four-group direct product of Z4 and Z4 and Z2 (1 copy) and direct product of Z8 and Z4 (3 copies) 21, 3 (direct product of Z4 and Z4 and Z2, direct product of Z8 and Z4) and (SmallGroup(32,2), semidirect product of Z8 and Z4 of M-type) (21,3) (1 copy) and (2,4) (1 copy)
cyclicity-preserving subgroup of second cohomology group for trivial group action cyclic group:Z2 direct product of Z4 and Z4 and Z2 and SmallGroup(32,2) 21, 2 (direct product of Z4 and Z4 and Z2 and SmallGroup(32,2)) (1 copy) and (direct product of Z8 and Z4 and semidirect product of Z8 and Z4 of M-type) (3 copies) (21,2) (1 copy) and (3,4) (2 copies)

Direct sum decomposition

For background information, see formula for second cohomology group for trivial group action of abelian group in terms of Schur multiplier and abelianization

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.

direct product of Z4 and Z4 and Z2 direct product of Z8 and Z4 direct product of Z8 and Z4 direct product of Z8 and Z4
SmallGroup(32,2) semidirect product of Z8 and Z4 of M-type semidirect product of Z8 and Z4 of M-type semidirect product of Z8 and Z4 of M-type

Generalized Baer Lie rings

The examples here illustrate the cocycle skew reversal generalization of Baer correspondence.

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.

direct product of Z4 and Z4 and Z2 direct product of Z8 and Z4 direct product of Z8 and Z4 direct product of Z8 and Z4
SmallGroup(32,2) semidirect product of Z8 and Z4 of M-type semidirect product of Z8 and Z4 of M-type semidirect product of Z8 and Z4 of M-type

Thus, we have two correspondences emerging:

Group GAP ID Additive group of Lie ring GAP ID More about the correspondence
SmallGroup(32,2) 2 direct product of Z4 and Z4 and Z2 21
semidirect product of Z8 and Z4 of M-type 4 direct product of Z8 and Z4 3

GAP implementation

Construction of the cohomology group

The cohomology group can be constucted using the GAP functions DirectProduct, CyclicGroup, TwoCohomology, TrivialGModule, GF.

gap> G := DirectProduct(CyclicGroup(4),CyclicGroup(4));;
gap> A := TrivialGModule(G,GF(2));;
gap> T := TwoCohomology(G,A);
rec( group := <pc group of size 16 with 4 generators>, module := rec( field := GF(2), isMTXModule := true, dimension := 1,
      generators := [ <an immutable 1x1 matrix over GF2>, <an immutable 1x1 matrix over GF2>, <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2> ] ),
  collector := rec( relators := [ [ [ 2, 1 ] ], [ [ 2, 1 ], 0 ], [ [ 3, 1 ], [ 3, 1 ], [ 4, 1 ] ], [ [ 4, 1 ], [ 4, 1 ], [ 4, 1 ], 0 ] ],
      orders := [ 2, 2, 2, 2 ], wstack := [ [ 2, 1 ], [ 2, 1 ] ], estack := [  ], pstack := [ 3, 3 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ],
      list := [ 1, 1, 0, 0 ], module := [ <an immutable 1x1 matrix over GF2>, <an immutable 1x1 matrix over GF2>, <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2> ], mone := <an immutable 1x1 matrix over GF2>, mzero := <an immutable 1x1 matrix over GF2>, avoid := [  ],
      unavoidable := [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] ), cohom := <linear mapping by matrix, <vector space of dimension 5 over GF(2)> -> ( GF(2)^3 )>,
  presentation := rec( group := <free group on the generators [ f1, f2, f3, f4 ]>, relators := [ f1^2*f2^-1, f1^-1*f2*f1*f2^-1, f2^2, f1^-1*f3*f1*f3^-1,
          f2^-1*f3*f2*f3^-1, f3^2*f4^-1, f1^-1*f4*f1*f4^-1, f2^-1*f4*f2*f4^-1, f3^-1*f4*f3*f4^-1, f4^2 ] ) )

Construction of extensions

The extensions can be constructed using the additional command Extensions.

gap> G := DirectProduct(CyclicGroup(4),CyclicGroup(4));;
gap> A := TrivialGModule(G,GF(2));;
gap> L := Extensions(G,A);;
gap> List(L,IdGroup);
[ [ 32, 21 ], [ 32, 3 ], [ 32, 2 ], [ 32, 4 ], [ 32, 3 ], [ 32, 3 ], [ 32, 4 ], [ 32, 4 ] ]

Under the action of the various automorphism groups

This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives.

gap> G := DirectProduct(CyclicGroup(4),CyclicGroup(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);
[ [ 32, 21 ], [ 32, 3 ], [ 32, 2 ], [ 32, 4 ] ]