Second cohomology group for trivial group action of S5 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 symmetric group:S5 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and symmetric group:S5 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 Klein four-group.
Get more specific information about symmetric group:S5 |Get more specific information about cyclic group:Z2|View other constructions whose value is Klein four-group

Description of the group

This article describes the second cohomology group for trivial group action:

\! H^2(G;A)

where G is symmetric group:S5 (the symmetric group of degree five) and A is cyclic group:Z2. Note that G has order 120 and A has order 2, so all the corresponding extensions have order 120 \times 2 = 240.

The cohomology group itself is isomorphic to the Klein four-group.

Computation of the group

The group can be computed using group cohomology of symmetric group:S5#Cohomology groups for trivial group action. As per this, we have:

H^2(G;A) \cong A/2A \oplus \operatorname{Ann}_A(2)

For A cyclic of order two, both A/2A and \operatorname{Ann}_A(2) are cyclic of order two.


Note that in all cases, the base of the group extension (isomorphic to cyclic group:Z2) equals the center of the group extension and hence is a characteristic subgroup.

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 240) Base characteristic in whole group?
trivial 1 direct product of S5 and Z2 189 Yes
nontrivial 1  ? 89 Yes
nontrivial 1  ? 90 Yes
nontrivial 1  ? 91 Yes