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 GeneralizedDihedralGroup 2013-03-22 14:53:28 2013-03-22 14:53:28 yark (2760) yark (2760) 9 yark (2760) Definition msc 20E22 generalised dihedral group DihedralGroup infinite dihedral group infinite dihedral