Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group/Proof 2

Theorem

Let $G$ be a group with the following properties:

$(1): \quad G$ is non-abelian.

$(2): \quad G$ is of order $8$.

$(3): \quad G$ has precisely one element of order $2$.

Then $G$ is isomorphic to the quaternion group $Q$.

Proof

By Groups of Order 8, the only group satisfying all three properties is $Q = \Dic 2$.

$\blacksquare$