Second cohomology group for trivial group action of V4 on direct product of Z4 and Z2
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 direct product of Z4 and Z2. The elements of this classify the group extensions with direct product of Z4 and Z2 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.
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Description of the group
where and .
The group is isomorphic to elementary abelian group:E64.
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)|
|trivial||1||direct product of E8 and Z4||45|
|one of the symmetric nontrivial ones||3||direct product of Z4 and Z4 and Z2||21|
|one of the symmetric nontrivial ones||6||direct product of Z8 and V4||36|
|one of the symmetric nontrivial ones||6||direct product of Z8 and Z4||3|
|non-symmetric||3||direct product of SmallGroup(16,13) and Z2||48|
|non-symmetric||9||direct product of M16 and Z2||37|
|non-symmetric||18||semidirect product of Z8 and Z4 of M-type||4|
Subgroups of interest
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
For this particular choice of and , the symmetric cohomology classes (corresponding to abelian group extensions) and the cyclicity-preserving subgroup generate the whole group, i.e., we have:
as an internal direct sum. In particular, every extension has a generalized Baer cyclicity-preserving Lie ring. A pictorial description of this would be as follows. Here, each column is a coset of and each row is a coset of . 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.
To avoid unnecessary duplication, we have compressed the table as follows: for each repeated column we have simply indicated the number of repeats in parentheses. The full table has 16 columns.