Second cohomology group for trivial group action of Z4 on V4

From Groupprops
Jump to: navigation, search

Description of the group

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

\! H^2(G,A)

where G \cong  \mathbb{Z}_4 and A \cong V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2.

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

Note that since cyclic over central implies abelian, all the corresponding group extensions are abelian. Equivalently, all the 2-cocycles are symmetric 2-cocycles.


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 16)
trivial 1 0 everywhere direct product of Z4 and V4 10
nontrivial 3 direct product of Z8 and Z2 5