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 to the free product of two cyclic groups of order .
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 .
|Title||generalized dihedral group|
|Date of creation||2013-03-22 14:53:28|
|Last modified on||2013-03-22 14:53:28|
|Last modified by||yark (2760)|
|Synonym||generalised dihedral group|
|Defines||infinite dihedral group|