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generalized dihedral group (Definition)

Let $ A$ be an abelian group. The generalized dihedral group $ {\mathrm{Dih}}(A)$ is the semidirect product $ A\rtimes C_2$, where $ C_2$ is the cyclic group of order $ 2$, and the 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_\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 $ {\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)$.



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See Also: dihedral group

Other names:  generalised dihedral group
Also defines:  infinite dihedral group, infinite dihedral
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Cross-references: abelian, complement, group, index, subgroup, non-Abelian, elementary abelian, free product, isomorphic, dihedral group, cyclic, inverses, maps, order, cyclic group, semidirect product, abelian group
There are 5 references to this entry.

This is version 6 of generalized dihedral group, born on 2004-12-13, modified 2006-03-19.
Object id is 6572, canonical name is GeneralizedDihedralGroup.
Accessed 5833 times total.

Classification:
AMS MSC: 20E22 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Extensions, wreath products, and other compositions)
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