Quaternion Group/Subgroup Generated by a^2/Quotient Group

Quotient Group of Subgroup Generated by $a^2$ of the Quaternion Group $Q$

Let the quaternion group $Q$ be represented by its group presentation:

$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$

Consider the subgroup $\gen {a^2}$ of $Q$:

$\gen {a^2} = \set {e, a^2}$

Let $E := N, A := a N, B := b N, C := a b N$.

Thus the quotient group of $G$ by $N$ is:

$G / N = \set {E, A, B, C}$

whose Cayley table can be presented as:

$\begin{array}{c|cccc}

 & E & A & B & C \\

\hline E & E & A & B & C \\ A & A & E & C & B \\ B & B & C & E & A \\ C & C & B & A & E \\ \end{array}$

which is seen to be an example of the Klein $4$-group.

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Exercise $3$