Supergroups of cyclic group:Z4

This article discusses some of the supergroups of the cyclic group of order four.

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:


 * 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 $$H^2(Q,N)$$.