Generalized dihedral group: Difference between revisions

Revision as of 21:43, 27 August 2009

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:

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

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

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 $n = \| H \|$ and $2^{k} = \| H / S \|$ where $S$ is the set of squares in $H$ .

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

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

Subgroups of $H$ : All of these are normal subgroups of $G$ . The number of such subgroups equals the number of subgroups of $H$ .
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 $G ∖ H$ . The number of such subgroups equals the sum of subgroup indices of $H$ .

@@ Line 45: / Line 45: @@
 | [[number of subgroups]] || [[number of subgroups]] of <math>H</math> plus [[sum of indices of subgroups]] of <math>H</math>. ||
 |-
-| [[number of conjugacy classes]] || <math>(n + 3 \cdot 2^k)/2</math> where <math>2^k</math> is the largest power of <math>2</math> dividing <math>n</math>. ||
+| [[number of conjugacy classes]] || <math>(n + 3 \cdot 2^k)/2</math> where <math>n = |H|</math> and <math>2^k = |H/S|</math> where <math>S</math> is the set of squares in <math>H</math>.
 |}