generalized dihedral group

Let $A$ be an abelian group. The generalized dihedral group $\mathrm{Dih}(A)$ is the semidirect product $A⋊C_2$, 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 $\mathrm{Dih}(A)$ is called a dihedral group. The finite dihedral group $\mathrm{Dih}(C_n)$ is commonly denoted by $D_n$ or $D_{2n}$ (the differing conventions being a source of confusion). The infinite dihedral group $\mathrm{Dih}(C_{\mathrm{\infty }})$ is denoted by $D_{\mathrm{\infty }}$, 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 $\mathrm{Dih}(A)$. If $A$ is not an elementary abelian $2$-group, then $\mathrm{Dih}(A)$ is non-abelian.

The subgroup $A\times \{1\}$ of $\mathrm{Dih}(A)$ is of index $2$, and every element of $\mathrm{Dih}(A)$ 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 $G\cong \mathrm{Dih}(N)$.

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