Second cohomology group for trivial group action of V4 on Z8

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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 Klein four-group on cyclic group:Z8. The elements of this classify the group extensions with cyclic group:Z8 in the center and Klein four-group the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
The value of this cohomology group is elementary abelian group:E8.
Get more specific information about Klein four-group |Get more specific information about cyclic group:Z8|View other constructions whose value is elementary abelian group:E8

Description of the group

We consider here the second cohomology group for trivial group action of the Klein four-group on cyclic group:Z8, i.e.,

H^2(G,A)

where G \cong V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2 and A \cong \mathbb{Z}_8.

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

Elements

FACTS TO CHECK AGAINST (second cohomology group for trivial group action):
Background reading on relationship with extension groups: Group extension problem
Arithmetic functions of extension group:
order (thus all extension groups have the same order): order of extension group is product of order of normal subgroup and quotient group
nilpotency class: nilpotency class of extension group is between nilpotency class of quotient group and one more for central extension
derived length: derived length of extension group is bounded by sum of derived length of normal subgroup and quotient group
minimum size of generating set: minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group|minimum size of generating set of quotient group is at most minimum size of generating set of group
WHAT'S THE TABLE BELOW?: Recall that there is a correspondence:
Elements of the group H^2(G;A) for the trivial group action \leftrightarrow congruence classes of central extensions with the specified subgroup A and quotient group G.
This descends to a correspondence:
Orbits for the group action of \operatorname{Aut}(G) \times \operatorname{Aut}(A) on H^2(G;A) \leftrightarrow pseudo-congruence classes of central extensions.
The table below breaks down the second cohomology group as a union of these orbits, with (as a general rule) each row describing one orbit, i.e., one "cohomology class type", aka one "pseudo-congruence class" of central extensions. The number of rows is the number of pseudo-congruence classes of central extensions.

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

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 32) Stem extension? Base characteristic in extension group? Nilpotency class of extension group Derived length of extension group Minimum size of generating set of extension group Subgroup information on base in whole group
trivial 1 direct product of Z8 and V4 36 No No 1 1 3
symmetric and nontrivial 3 direct product of Z16 and Z2 16 No Yes 1 1 2
non-symmetric 3 M32 17 No Yes 2 2 2
non-symmetric 1 central product of D8 and Z8 38 No Yes 2 2 3
Total (--) 8 -- -- -- -- -- -- -- --

Group actions

Under the action of the automorphism group of the Klein four-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 Z16 and Z2 are in one orbit, while the 3 cohomology classes that give M32 are in another orbit.

Under the action of the automorphism group of the cyclic four-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.

Subgroups of interest

Subgroup Value Corresponding group extensions for subgroup GAP IDs Group extension groupings for each coset GAP IDs
IIP subgroup of second cohomology group for trivial group action cyclic group:Z2 direct product of Z8 and V4 (1 copy) and central product of D8 and Z8 (1 copy) 10, 13 (direct product of Z8 and V4, central product of D8 and Z8) (1 copy) and (direct product of Z16 and Z2, M32) (3 copies) (36,38) (1 copy) and (16,17) (3 copies)
cyclicity-preserving subgroup of second cohomology group for trivial group action cyclic group:Z2 direct product of Z8 and V4 (1 copy) and central product of D8 and Z8 (1 copy) 10, 13 (direct product of Z8 and V4, central product of D8 and Z8) (1 copy) and (direct product of Z16 and Z2, M32) (3 copies) (36,38) (1 copy) and (16,17) (3 copies)
subgroup generated by images of symmetric 2-cocycles (corresponds to abelian group extensions) Klein four-group direct product of Z8 and V4 (1 copy) and direct product of Z16 and Z2 (3 copies) 10, 5 (direct product of Z8 and V4, direct product of Z16 and Z2) and (central product of D8 and Z8, M32) (36,38) (1 copy) and (16,17) (1 copy)

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. 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.

Generalized Baer Lie rings

The examples here illustrate the cocycle halving generalization of Baer correspondence. 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

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 Z8 and V4 direct product of Z16 and Z2 direct product of Z16 and Z2 direct product of Z16 and Z2
central product of D8 and Z8 M32 M32 M32

Thus, we have two correspondences emerging:

Group GAP ID Additive group of Lie ring GAP ID More about the correspondence
central product of D8 and Z8 (32,38) direct product of Z8 and V4 (32,36) generalized Baer correspondence between central product of D8 and Z8 and direct product of Z8 and V4
M32 (32,17) direct product of Z16 and Z2 (32,16) generalized Baer correspondence between M32 and direct product of Z16 and Z2