Quaternion Group is Hamiltonian

Theorem
The quaternion group $Q$ is Hamiltonian.

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

By definition $Q$ is Hamiltonian :
 * $Q$ is non-abelian

and:
 * every subgroup of $Q$ is normal.

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


 * $a b \ne b a$

From Subgroups of Quaternion Group:

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.