Second cohomology group for trivial group action of Z2 on Z4

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

Description of the group

We consider here the second cohomology group for trivial group action of the cyclic group:Z2 on the cyclic group:Z4, i.e.,

\! H^2(G,A)

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

The cohomology group is isomorphic to cyclic group:Z2.

Elements

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 Representative 2-cocycle Corresponding group extension Second part of GAP ID (order is 8)
trivial 1 0 everywhere direct product of Z4 and Z2 2
nontrivial 1 f(1,1) = 1, all others 0 cyclic group:Z8 1