Conjugacy Classes of Quaternion Group

Theorem
Let $Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the quaternion group.

The conjugacy classes of $\Dic 2$ are:
 * $\set e, \set {a^2}, \set {a, a^3}, \set {b, a^2 b}, \set {a b, a^3 b}$

Proof
From Center of Quaternion Group, we have:
 * $\map Z {\Dic 2} = \set {e, a^2}$

Thus from Conjugacy Class of Element of Center is Singleton, $\set e$ and $\set {a^2}$ are two of those conjugacy classes.

By inspection of the Cayley table:

we investigate the remaining $6$ elements in turn, starting with $a$:

So we have a conjugacy class:
 * $\set {a, a^3}$

Investigating the remaining $4$ elements in turn, starting with $b$:

So we have a conjugacy class:
 * $\set {b, a^2 b}$

Investigating the remaining $2$ elements, starting with $a b$:

We need go no further: the remaining elements $a b$ and $a^3 b$ are in the same conjugacy class:
 * $\set {a b, a^3 b}$

Hence the result.