Generalized dihedral group

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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

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:

  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.