generalized dihedral group

Let $A$ be an abelian group. The generalized dihedral group $\operatorname{Dih}(A)$ is the semidirect product $A\rtimes C_{2}$ , where $C_{2}$ is the cyclic group of order $2$ , and the generator (http://planetmath.org/Generator) of $C_{2}$ maps elements of $A$ to their inverses.

If $A$ is cyclic, then $\operatorname{Dih}(A)$ is called a dihedral group. The finite dihedral group $\operatorname{Dih}(C_{n})$ is commonly denoted by $D_{n}$ or $D_{2n}$ (the differing conventions being a source of confusion). The infinite dihedral group $\operatorname{Dih}(C_{\infty})$ is denoted by $D_{\infty}$ , and is isomorphic to the free product $C_{2}*C_{2}$ of two cyclic groups of order $2$ .

If $A$ is an elementary abelian $2$ -group, then so is $\operatorname{Dih}(A)$ . If $A$ is not an elementary abelian $2$ -group, then $\operatorname{Dih}(A)$ is non-abelian.

The subgroup $A\times\{1\}$ of $\operatorname{Dih}(A)$ is of index $2$ , and every element of $\operatorname{Dih}(A)$ 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 $G\setminus N$ are of order $2$ , then $N$ is abelian and $G\cong\operatorname{Dih}(N)$ .

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