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

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 Z2 on cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 in the center and direct product of Z4 and Z2 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 Z2 |Get more specific information about cyclic group:Z4

Contents

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.

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 32) Stem extension? Base characteristic in whole group Nilpotency class of whole group (at least 1, at most 2) Derived length of whole group (at least 1, at most 2) Minimum size of generating set of whole group (at least 2, at most 3)
trivial 1 direct product of Z4 and Z4 and Z2 21 No No 1 1 3
symmetric and nontrivial 4 direct product of Z16 and Z2 16 No Yes 1 1 2
symmetric and nontrivial 1 direct product of Z8 and V4 36 No No 1 1 3
symmetric and nontrivial 2 direct product of Z8 and Z4 3 No Yes 1 1 2
non-symmetric 1 SmallGroup(32,24) 24 No No 2 2 3
non-symmetric 4 M32 17 No Yes 2 2 2
non-symmetric 1 direct product of M16 and Z2 37 No Yes 2 2 3
non-symmetric 2 semidirect product of Z8 and Z4 of M-type 4 No No 2 2 2

Subgroups of interest

Subgroup Value Corresponding group extensions for subgroup GAP IDs second part Group extension groupings for each coset GAP IDs second part
cyclicity-preserving subgroup of second cohomology group for trivial group action cyclic group:Z2 direct product of Z4 and Z4 and Z2, SmallGroup(32,24) 21, 24 (direct product of Z4 and Z4 and Z2, SmallGroup(32,24)) (1 copy), (direct product of Z16 and Z2, M32) (4 copies), (direct product of Z8 and V4, direct product of M16 and Z2 (1 copy), (direct product of Z8 and Z4, semidirect product of Z8 and Z4 of M-type) (2 copies) (21, 24) (1 copy), (16, 17) (2 copies), (3, 4) (4 copies), (36, 37) (1 copy)
subgroup of second cohomology group comprising symmetric 2-cocycles, corresponds to abelian group extensions direct product of Z4 and Z2 direct product of Z4 and Z4 and Z2 (1 copy), direct product of Z16 and Z2 (4 copies), direct product of Z8 and V4 (1 copy), direct product of Z8 and Z4 (2 copies) 21,16,36,3 (direct product of Z4 and Z4 and Z2 (1 copy), direct product of Z16 and Z2 (4 copies), direct product of Z8 and Z4 (2 copies), direct product of Z8 and V4 (1 copy)), (SmallGroup(32,24) (1 copy), M32 (4 copies), direct product of M16 and Z2 (1 copy), semidirect product of Z8 and Z4 of M-type (2 copies)) (21, 16, 36, 3), (24, 17, 37, 4)

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

See also 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

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 Z16 and Z2 direct product of Z16 and Z2 direct product of Z16 and Z2 direct product of Z16 and Z2 direct product of Z8 and V4 direct product of Z8 and Z4 direct product of Z8 and Z4
SmallGroup(32,24) M32 M32 M32 M32 direct product of M16 and Z2 semidirect product of Z8 and Z4 of M-type semidirect product of Z8 and Z4 of M-type

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

Generalized Baer Lie rings

The examples here illustrate the cocycle halving 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 Z16 and Z2 direct product of Z16 and Z2 direct product of Z16 and Z2 direct product of Z16 and Z2 direct product of Z8 and V4 direct product of Z8 and Z4 direct product of Z8 and Z4
SmallGroup(32,24) M32 M32 M32 M32 direct product of M16 and Z2 semidirect product of Z8 and Z4 of M-type semidirect product of Z8 and Z4 of M-type

We thus get the following correspondences:

Group GAP ID Additive group of Lie ring GAP ID More about the correspondence
SmallGroup(32,24) 24 direct product of Z4 and Z4 and Z2 21 generalized Baer correspondence between SmallGroup(32,24) and direct product of Z4 and Z4 and Z2
M32 17 direct product of Z16 and Z2 16 generalized Baer correspondence between M32 and direct product of Z16 and Z2
direct product of M16 and Z2 37 direct product of Z8 and V4 36 generalized Baer correspondence between direct product of M16 and Z2 and direct product of Z8 and V4
semidirect product of Z8 and Z4 of M-type 4 direct product of Z8 and Z4 3 generalized Baer correspondence between nontrivial semidirect product of Z8 and Z4 of M-type and direct product of Z8 and Z4