Conjugacy Classes of the Dihedral Group, D4

Conjugacy Classes of the Dihedral Group, D4

Let $D_4 = \langle r, s : r^4 = s^2 = 1, (rs)^2 = 1 \rangle = \{ 1, r, r^2, r^3, s, sr, sr^2, sr^3 \}$ where $r$ denotes the counterclockwise rotation translation, and $s$ denotes the flip translation.

The multiplication table for $D_4$ is given below:

$1$ $r$ $r^2$ $r^3$ $s$ $rs$ $r^2s$ $r^3s$
$1$ $1$ $r$ $r^2$ $r^3$ $s$ $rs$ $r^2s$ $r^3s$
$r$ $r$ $r^2$ $r^3$ $1$ $rs$ $r^2s$ $r^3s$ $s$
$r^2$ $r^2$ $r^3$ $1$ $r$ $r^2s$ $r^3s$ $s$ $rs$
$r^3$ $r^3$ $1$ $r$ $r^2$ $r^3s$ $s$ $rs$ $r^2s$
$s$ $s$ $r^3s$ $r^2s$ $rs$ $1$ $r^3$ $r^2$ $r$
$rs$ $rs$ $s$ $r^3s$ $r^2s$ $r$ $1$ $r^3$ $r^2$
$r^2s$ $r^2s$ $rs$ $s$ $r^3s$ $r^2$ $r$ $1$ $r^3$
$r^3s$ $r^3s$ $r^2s$ $rs$ $s$ $r^3$ $r^2$ $r$ $1$

From this table we immediately see that $Z(D_4) = \{ 1, r^2 \}$. Each element of $Z(D_4)$ has a trivial conjugacy class, i.e., $[1] = \{ 1 \}$ and $[r^2] = \{ r^2 \}$.

Consider the element $r$. We have that:

(1)
\begin{align} \quad rrr^{-1} &= r \\ \quad r^3r(r^3)^{-1} &= r^3rr = r^5 = r \\ \quad srs^{-1} &= srs = r^3 \\ \quad (rs)r(rs)^{-1} &= (rs)r(s^{-1}r^{-1}) = (rs)(rs)r^3 = 1r^3 = r^3 \\ \quad (r^2s)r(r^2s)^{-1} &= (r^2s)r(s^{-1}r^{-2}) = (r^2s)(rs)r^2 = rr^2 = r^3 \\ \quad (r^3s)r(r^3s)^{-1} &= (r^3s)r(s^{-1}r^{-3}) = r^3srsr = (r^3s)(rs)r= r^2r = r^3 \end{align}

Thus the conjugacy class of $r$ is $[r] = \{ r, r^3 \}$.

Now consider the element $s$. We have that:

(2)
\begin{align} \quad sss^{-1} &= s \\ \quad (rs)s(rs)^{-1} &= (rs)s(s^{-1}r^3) = (rs)(r^3) = r^2s \\ \quad (r^2s)s(r^2s)^{-1} &= (r^2s)s(s^{-1}r^2) = (r^2s)(r^2) = s \\ \quad (r^3s)s(r^3s)^{-1} &= (r^3s)s(s^{-1}r) = (r^3s)(r) = r^2s \end{align}

Thus the conjugacy class of $s$ is $[s] = \{ s, r^2s \}$. There are only two remaining elements in $D_4$ that have not been assigned a conjugacy class yet. They are $rs$ and $r^3s$. Since $rs, r^3s \not \in Z(D_4)$ we have that $rs$ and $r^3s$ have nontrivial conjugacy classes. So $[rs] = \{ rs, r^3s \}$.

We can partition $D_4$ into its conjugacy classes by:

(3)
\begin{align} \quad D_4 = [1] \cup [r^2] \cup [r] \cup [s] \cup [rs] = \{ 1 \} \cup \{ r^2 \} \cup \{ r, r^3 \} \cup \{ s, r^2s \} \cup \{ rs, r^3s \} \end{align}

And The Class Equation of $D_4$ is:

(4)
\begin{align} \quad |D_4| = 2 + (2 + 2 + 2) \end{align}
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