# Quaternion Group is Hamiltonian

## Theorem

The quaternion group $Q$ is Hamiltonian.

## Proof

For clarity the Cayley table of $Q$ is presented below:

- $\begin{array}{r|rrrrrrrr} & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ \hline e & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ a & a & a^2 & a^3 & e & a b & a^2 b & a^3 b & b \\ a^2 & a^2 & a^3 & e & a & a^2 b & a^3 b & b & a b \\ a^3 & a^3 & e & a & a^2 & a^3 b & b & a b & a^2 b \\ b & b & a^3 b & a^2 b & a b & a^2 & a & e & a^3 \\ a b & a b & b & a^3 b & a^2 b & a^3 & a^2 & a & e \\ a^2 b & a^2 b & a b & b & a^3 b & e & a^3 & a^2 & a \\ a^3 b & a^3 b & a^2 b & a b & b & a & e & a^3 & a^2 \end{array}$

By definition $Q$ is Hamiltonian if and only if:

- $Q$ is non-abelian

and:

$Q$ is non-abelian as demonstrated by the counter-example:

- $a b \ne b a$

From Subgroups of Quaternion Group:

The subsets of $Q$ which form subgroups of $Q$ are:

\(\displaystyle \) | \(\) | \(\displaystyle Q\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set e\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set {e, a^2}\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set {e, a, a^2, a^3}\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set {e, b, a^2, a^2 b}\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set {e, a b, a^2, a^3 b}\) |

From Quaternion Group is Hamiltonian we have that all of these subgroups of $Q$ are normal.

From Trivial Subgroup and Group Itself are Normal:

- $Q$ and $\set e$ are normal subgroups of $Q$.

From Center of Quaternion Group, $\gen {a^2} = \set {e, a^2}$ is the center of $Q$.

From Center of Group is Normal Subgroup, $\set {e, a^2}$ is normal in $Q$.

The remaining subgroups of $Q$ are of order $4$, and so have index $2$.

From Subgroup of Index 2 is Normal it follows that all of these order $4$ subgroups of $Q$ are normal.

That accounts for all subgroups of $Q$.

Hence the result.

$\blacksquare$

## Sources

- 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.10$: Example $36$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 46 \iota$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $8 \ \text{(ii)}$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Exercise $3$