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.

Relation with other properties

Stronger properties

• Dihedral group
• Elementary abelian 2-group

Weaker properties

• Metabelian group
• Solvable 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 $L\leq H$ 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 $H$.