Dicyclic group

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

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with dihedral 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 dicyclic group, also called the binary dihedral group with parameter $n$ is defined in the following equivalent ways:

• It is given by the presentation:

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

• It has the following representation as a subgroup of the quaternions: $a=e^{i\pi /n},x=j$

The dicyclic group with parameter $n$ has order $4n$, and it is an extension of a cyclic group of order $2n$ by a cyclic group of order 2.

Particular cases

• When $n=2$ we get the quaternion group
• When $n$ is a power of $2$ we get a generalized quaternion group