Definition:Dicyclic Group

Definition

For even $n$, the dicyclic group $Dic_n$ of order $4 n$ is the group having the presentation:

$Dic_n = \left \langle{x, y: x^{2 n} = e, y^2 = x^n, y^{-1} x y = x^{-1}}\right \rangle$

Quaternion Group

The dicyclic group $Dic_2$ is known as the quaternion group.

The elements of $Dic_2$ are:

$Dic_2 = \left\{{e, a, a^2, a^3, b, a b, a^2 b, a^3 b}\right\}$

Its group presentation is given by:

$Dic_2 = \left \langle {a, b: a^4 = e, b^2 = a^2, a b a = b}\right \rangle$

Its Cayley table is given by:

$\begin{array}{c|cccccccc} & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ \hline e & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ a & a & a^2 & a^3 & e & a b & a^2 b & a^3 b & b \\ a^2 & a^2 & a^3 & e & a & a^2 b & a^3 b & b & a b \\ a^3 & a^3 & e & a & a^2 & a^3 b & b & a b & a^2 b \\ b & b & a^3 b & a^2 b & a b & a^2 & a & e & a^3 \\ a b & a b & b & a^3 b & a^2 b & a^3 & a^2 & a & e \\ a^2 b & a^2 b & a b & b & a^3 b & e & a^3 & a^2 & a \\ a^3 b & a^3 b & a^2 b & a b & b & a & e & a^3 & a^2 \end{array}$

Many sources (including this website) tend to refer to this group merely as $Q$.

Alternative Notation

Some sources denote the group $Dic_n$ as $Q_{2n}$, referring to it as the generalized quaternion group:

$Q_{2n} = \left \langle{x, y: x^{2n} = e, y^2 = x^n, y^{-1}xy = x^{-1} }\right \rangle$

Using this notation, it can be seen that the quaternion group is represented by:

$Q_4 = \left \langle{x, y: x^4 = e, y^2 = x^2, y^{-1}xy = x^{-1} }\right \rangle$

Others have a different notation again:

$Q_{4n} = \left \langle{x, y: x^{2n} = e, y^2 = x^n, y^{-1}xy = x^{-1} }\right \rangle$

Using this notation, it can be seen that the quaternion group is represented by:

$Q_8 = \left \langle{x, y: x^4 = e, y^2 = x^2, y^{-1}xy = x^{-1} }\right \rangle$

Because of the potential ambiguity, it is recommended that $Q_{2n}$ and $Q_{4n}$ are not used, but that $Dic_n$ is used throughout.

Also see

• Results about dicyclic groups can be found here.