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.

Arithmetic functions

Function Value Explanation
order Twice the order of $H$
exponent Least common multiple of $2$ and the exponent of $H$
derived length $1$ if $H$ is an elementary abelian $2$-group, $2$ otherwise.
nilpotency class Frattini length of $H$ if $H$ is a $2$-group, not defined otherwise.
max-length One more than the max-length of $H$.
composition length One more than the composition length of $H$.
chief length One more than the chief length of $H$.
minimum size of generating set One more than the minimum size of generating set of $H$.
number of subgroups number of subgroups of $H$ plus sum of indices of subgroups of $H$.
number of conjugacy classes $(n + 3 \cdot 2^k)/2$ where $n = |H|$ and $2^k = |H/S|$ where $S$ is the set of squares in $H$.

Group properties

Property Satisfied Explanation
Abelian group True only if $H$ is an elementary abelian $2$-group
Nilpotent group True only if $H$ is a $2$-group
Solvable group Yes
Metabelian group Yes

Subgroups

Further information: Subgroup structure of generalized dihedral groups

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 $H$.