Supergroups of cyclic group:Z4
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
The general procedure is as follows:
- Determine all homomorphisms from to .
- For each of these automorphisms, find all possible extensions, classified by the second cohomology group for that action.
is cyclic of order two, so there are two possible homomorphisms from to : the trivial homomorphism and the unique isomorphism.
The groups corresponding to the trivial homomorphism are:
- Direct product of Z4 and Z2: This corresponds to the identity element in .
- Cyclic group:Z8: This corresponds to the non-identity element in .
The groups corresponding to the unique isomorphism are: