Dihedral group

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with dicyclic group

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This is a family of groups parametrized by the natural numbers, viz, for each natural number, there is a unique group (upto isomorphism) in the family corresponding to the natural number. The natural number is termed the parameter for the group family

Definition

The dihedral group with parameter $n$, denoted sometimes as $D_{n}$ and sometimes as $D_{2n}$ is defined in the following equivalent ways:

• It has the presentation:

$<x,a|a^{n}=x^{2}=1,xax^{-1}=a^{-1}>$

• It is the group of symmetries of a regular $n$-gon in the plane, viz the plane isometries that preserves the set of points of the regular $n$-gon.