Generalized dihedral group: Difference between revisions

Latest revision as of 16:55, 17 February 2012

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:

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

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

Relation with other properties

Stronger properties

Weaker properties

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 $H$ :

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 $H$ : 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 $g L$ where $g \in G ∖ H$ , i.e., $g \in G, g \notin H$ . Thus, to specify $K$ , it suffices to specify $L$ and the coset $g L$ . Conversely, given any subgroup $L \leq H$ and any coset $g L$ with $g \in G ∖ H$ , we obtain a subgroup $K = L \cup g L$ . 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 $H$ .

@@ Line 21: / Line 21: @@
 * [[Stronger than::Metabelian group]]
 * [[Stronger than::Solvable group]]
+==Arithmetic functions==
+{| class="wikitable" border="1"
+! Function !! Value !! Explanation
+|-
+| [[Order of a group|order]] || Twice the order of <math>H</math> ||
+|-
+| [[Exponent of a group|exponent]] || Least common multiple of <math>2</math> and the exponent of <math>H</math> ||
+|-
+| [[derived length]] || <math>1</math> if <math>H</math> is an elementary abelian <math>2</math>-group, <math>2</math> otherwise. ||
+|-
+| [[nilpotency class]] || [[Frattini length]] of <math>H</math> if <math>H</math> is a <math>2</math>-group, not defined otherwise. ||
+|-
+| [[max-length]] || One more than the max-length of <math>H</math>. ||
+|-
+| [[composition length]] || One more than the composition length of <math>H</math>. ||
+|-
+| [[chief length]] || One more than the chief length of <math>H</math>. ||
+|-
+| [[minimum size of generating set]] || One more than the minimum size of generating set of <math>H</math>. ||
+|-
+| [[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>n = |H|</math> and <math>2^k = |H/S|</math> where <math>S</math> is the set of squares in <math>H</math>.
+|}
+==Group properties==
+{| class="wikitable" border="1"
+!Property !! Satisfied !! Explanation
+|-
+|[[Abelian group]] || True only if <math>H</math> is an elementary abelian <math>2</math>-group ||
+|-
+|[[Nilpotent group]] || True only if <math>H</math> is a <math>2</math>-group ||
+|-
+|[[Solvable group]] || Yes ||
+|-
+|[[Metabelian group]] || Yes ||
+|}
 ==Subgroups==
@@ Line 29: / Line 69: @@
 # Subgroups of <math>H</math>: All of these are [[normal subgroup]]s of <math>G</math>. The number of such subgroups equals the [[number of subgroups]] of <math>H</math>.
-# Subgroups of <math>G</math> containing an element outside <math>H</math>: These are classified by the following two pieces of information: the choice of a subgroup of <math>H</math>, and the choice of a coset of that subgroup in <math>G \setminus H</math>. The number of such subgroups equals the [[sum of subgroup indices]] of <math>H</math>.
+# Subgroups of <math>G</math> containing an element outside <math>H</math>: Suppose <math>K</math> is such a subgroup. Then <math>L =H \cap K</math> is a [[subgroup of index two]] in <math>K</math>, and <math>K</math> is the union of <math>L</math> and a coset <math>gL</math> where <math>g \in G \setminus H</math>, i.e., <math>g \in G, g \notin H</math>. Thus, to specify <math>K</math>, it suffices to specify <math>L</math> and the coset <math>gL</math>. Conversely, given ''any'' subgroup <math>L \le H</math> and ''any'' coset <math>gL</math> with <math>g \in G \setminus H</math>, we obtain a subgroup <math>K = L \cup gL</math>. The reason this is a subgroup is because <math>g</math> has order two and acts on <math>L</math> by the inverse map. The upshot is that, given <math>L</math>, the number of possibilities for <math>K</math> is the number of cosets of <math>L</math> outside <math>H</math>, which equals the number of cosets inside <math>H</math>, which equals the index <math>[H:L]</math>. Thus, the ''total'' number of possibilities for <math>K</math> is the [[sum of subgroup indices]] over all subgroups of <math>H</math>.