generalized dihedral group
Let be an abelian group.The generalized dihedral group is the semidirect product
,where is the cyclic group
of order ,and the generator
(http://planetmath.org/Generator) of maps elements of to their inverses
.
If is cyclic, then is called a dihedral group.The finite dihedral group is commonly denoted by or (the differing conventions being a source of confusion).The infinite dihedral group is denoted by ,and is isomorphic
tothe free product
of two cyclic groups of order .
If is an elementary abelian -group, then so is .If is not an elementary abelian -group, then is non-abelian.
The subgroup of is of index ,and every element of that is not in this subgroup has order .This property in fact characterizes generalized dihedral groups,in the sense that if a group has a subgroup of index such that all elements of the complement are of order ,then is abelian and .