# Conjugacy Classes of the Quaternion Group, Q8

Let $Q_8 = \{ 1, -1, i, \bar{i}, j, \bar{j}, k, \bar{k} \}$ be The Quaternion Group, Q8 (see link for the group operation table). We will described all of the conjugacy classes of $Q_8$.

We first look at $Z(Q_8)$. Recall that $ij = k \neq \bar{k} = \neq ji$, $jk = i \neq \bar{i} =\neq kj$, and $ki = j \neq \bar{j} = ik$. Thus $i, j, k \not \in Z(Q_8)$. Similarly $\bar{i}j \neq j \bar{i}$, $\bar{j}k \neq k\bar{j}$, and $\bar{k}i \neq i\bar{k}$, so $\bar{i}, \bar{j}, \bar{k} \not \in Z(Q_8)$. We see that only $1, -1 \in Z(Q_8)$.

Recall that $g \in G$ has a trivial conjugacy class if and only if $g \in Z(G)$. Thus we see that the conjugacy classes of $1$ and $-1$ are trivial, that is:

(1)Now look at the element $i \in Q_8$. We see that:

(2)Therefore the conjugacy class of $i$ is $[i] = \{ i, \bar{i} \}$. It can similarly be shown that the conjugacy classes of $[j]$ and $[k]$ are as expected, $[j] = \{j, \bar{j} \}$ and $[k] = \{k, \bar{k} \}$. Thus, $Q_8$ is partioned into its conjugacy classes by:

(3)Moreover, The Class Equation for $Q_8$ is:

(4)