Generalized dihedral group
Suppose is an abelian group. The generalized dihedral group corresponding to is the external semidirect product of with the cyclic group of order two, with the non-identity element acting as the inverse map on .
A presentation for is:
Relation with other properties
|order||Twice the order of|
|exponent||Least common multiple of and the exponent of|
|derived length||if is an elementary abelian -group, otherwise.|
|nilpotency class||Frattini length of if is a -group, not defined otherwise.|
|max-length||One more than the max-length of .|
|composition length||One more than the composition length of .|
|chief length||One more than the chief length of .|
|minimum size of generating set||One more than the minimum size of generating set of .|
|number of subgroups||number of subgroups of plus sum of indices of subgroups of .|
|number of conjugacy classes||where and where is the set of squares in .|
|Abelian group||True only if is an elementary abelian -group|
|Nilpotent group||True only if is a -group|
Further information: Subgroup structure of generalized dihedral groups
There are two kinds of subgroups of the generalized dihedral group with the abelian subgroup :
- Subgroups of : All of these are normal subgroups of . The number of such subgroups equals the number of subgroups of .
- Subgroups of containing an element outside : Suppose is such a subgroup. Then is a subgroup of index two in , and is the union of and a coset where , i.e., . Thus, to specify , it suffices to specify and the coset . Conversely, given any subgroup and any coset with , we obtain a subgroup . The reason this is a subgroup is because has order two and acts on by the inverse map. The upshot is that, given , the number of possibilities for is the number of cosets of outside , which equals the number of cosets inside , which equals the index . Thus, the total number of possibilities for is the sum of subgroup indices over all subgroups of .