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:

$G:=⟨H,x\mid xh{x}^{-1}=h^{-1}\mathrm{\forall }h\in H⟩$.

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 $H$
Dderived 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 $n=\left|H\right|$ 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 $H$: These are classified by the following two pieces of information: the choice of a subgroup of $H$, and the choice of a coset of that subgroup in $G\setminus H$. The number of such subgroups equals the sum of subgroup indices of $H$.