Supergroups of cyclic group:Z4

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This article gives specific information, namely, supergroups, about a particular group, namely: cyclic group:Z4.
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This article discusses some of the supergroups of the cyclic group of order four.

Classification of supergroups of order eight

Further information: Classification of finite p-groups with cyclic maximal subgroup

The general procedure

We classify groups G of order eight containing the cyclic group of order four as a normal subgroup N. The quotient group G/N must therefore be isomorphic to the cyclic group of order two.

The general procedure is as follows:

  • Determine all homomorphisms from Q to \operatorname{Aut}(N).
  • For each of these automorphisms, find all possible extensions, classified by the second cohomology group H^2(Q,N) for that action.

The classification

\operatorname{Aut}(N) is cyclic of order two, so there are two possible homomorphisms from Q to \operatorname{Aut}(N): the trivial homomorphism and the unique isomorphism.

The groups corresponding to the trivial homomorphism are:

The groups corresponding to the unique isomorphism are: