Generalized dihedral group

Definition
Suppose $$H$$ is an abelian group. The generalized dihedral group corresponding to $$H$$ is the external semidirect product of $$H$$ with the cyclic group of order two, with the non-identity element acting as the inverse map on $$H$$.

Viewing this external semidirect product as an internal semidirect product, $$H$$ is an abelian normal subgroup of index two.

A presentation for $$G$$ is:

$$G := \langle H,x \mid xhx^{-1} = h^{-1} \forall h \in H \rangle$$.

Note that the dihedral groups are special cases of generalized dihedral groups where the abelian group in question is a cyclic group.

Stronger properties

 * Weaker than::Dihedral group
 * Weaker than::Elementary abelian 2-group

Weaker properties

 * Stronger than::Metabelian group
 * Stronger than::Solvable group

Subgroups
There are two kinds of subgroups of the generalized dihedral group $$G$$ with the abelian subgroup $$H$$:


 * 1) Subgroups of $$H$$: All of these are normal subgroups of $$G$$. The number of such subgroups equals the number of subgroups of $$H$$.
 * 2) Subgroups of $$G$$ containing an element outside $$H$$: Suppose $$K$$ is such a subgroup. Then $$L =H \cap K$$ is a subgroup of index two in $$K$$, and $$K$$ is the union of $$L$$ and a coset $$gL$$ where $$g \in G \setminus H$$, i.e., $$g \in G, g \notin H$$. Thus, to specify $$K$$, it suffices to specify $$L$$ and the coset $$gL$$. Conversely, given any subgroup $$L \le H$$ and any coset $$gL$$ with $$g \in G \setminus H$$, we obtain a subgroup $$K = L \cup gL$$. The reason this is a subgroup is because $$g$$ has order two and acts on $$L$$ by the inverse map. The upshot is that, given $$L$$, the number of possibilities for $$K$$ is the number of cosets of $$L$$ outside $$H$$, which equals the number of cosets inside $$H$$, which equals the index $$[H:L]$$. Thus, the total number of possibilities for $$K$$ is the sum of subgroup indices over all subgroups of $$H$$.