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

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 $\mathrm{A}\mathrm{u}\mathrm{t}(N)$.
• For each of these automorphisms, find all possible extensions, classified by the second cohomology group $H^2(Q,N)$ for that action.

The classification

$\mathrm{A}\mathrm{u}\mathrm{t}(N)$ is cyclic of order two, so there are two possible homomorphisms from $Q$ to $\mathrm{A}\mathrm{u}\mathrm{t}(N)$: 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 $H^2(Q,N)$.
• Cyclic group:Z8: This corresponds to the non-identity element in $H^2(Q,N)$.

The groups corresponding to the unique isomorphism are:

• Dihedral group:D8: This corresponds to the identity element in $H^2(Q,N)$.
• Quaternion group: This corresponds to the non-identity element in $H^2(Q,N)$.