generalized dihedral group

Let A be an abelian groupMathworldPlanetmath. The generalized dihedral group Dih(A) is the semidirect productMathworldPlanetmath AC2, where C2 is the cyclic groupMathworldPlanetmath of order 2, and the generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( of C2 maps elements of A to their inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

If A is cyclic, then Dih(A) is called a dihedral groupMathworldPlanetmath. The finite dihedral group Dih(Cn) is commonly denoted by Dn or D2n (the differing conventions being a source of confusion). The infinite dihedral group Dih(C) is denoted by D, and is isomorphicPlanetmathPlanetmathPlanetmath to the free productMathworldPlanetmath C2*C2 of two cyclic groups of order 2.

If A is an elementary abelian 2-group, then so is Dih(A). If A is not an elementary abelian 2-group, then Dih(A) is non-abelianMathworldPlanetmathPlanetmath.

The subgroupMathworldPlanetmathPlanetmath A×{1} of Dih(A) is of index 2, and every element of Dih(A) that is not in this subgroup has order 2. This property in fact characterizes generalized dihedral groups, in the sense that if a group G has a subgroup N of index 2 such that all elements of the complement GN are of order 2, then N is abelian and GDih(N).

Title generalized dihedral group
Canonical name GeneralizedDihedralGroup
Date of creation 2013-03-22 14:53:28
Last modified on 2013-03-22 14:53:28
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Definition
Classification msc 20E22
Synonym generalised dihedral group
Related topic DihedralGroup
Defines infinite dihedral group
Defines infinite dihedral