What are the Conjugacy Classes of D3?

The conjugacy classes of

D_3

are

\{e\}

\{r,r^2\}

and

\{s,sr,sr^2\}

. We know that

\begin{equation*} D_3 = \langle r,s \ | \ r^3 = s^2 = e, rs = sr^{-1} \rangle \end{equation*}

Some books may write it as

D_6

, but we prefer to take the

D_n

instead of

D_{2n}

. To determine the conjugates of the element

\tau \in D_3

, we need to show determine

\sigma \tau \sigma^{-1}

for all

\sigma \in D_3

. So we will check one by one the conjugates of the elements of the dihedral group

D_3

: $\tau = e$

\begin{align*} eee &= e \\ rer^{-1} &= e \\ r^2er^{-2} &= e \\ ses^{-1} &= e \\ rse(rs)^{-1} &= e \\ r^2se(r^2s)^{-1} &= e \\ \end{align*}

$\tau = r$

\begin{align*} ere &= r \\ rrr^{-1} &= r \\ r^2rr^{-2} &= r^{2 + 1 - 2} = r \\ srs^{-1} &= srs = ssr^{-1} = r^{2} \\ rsr(rs)^{-1} &= rsrs^{-1}r^{-1} = sr^{-1}rs^{-1}r^{-1} = r^{-1} = r^{2} \\ r^2sr(r^2s)^{-1} &= rsr^{-1}rs^{-1}r^{-2} = r^{-1} = r^{2} \\ \end{align*}

$\tau = r^2$

\begin{align*} er^2e &= r^2 \\ rr^2r^{-1} &= r^2 \\ r^2r^2r^{-2} &= r^{2 + 2 - 2} = r^2 \\ sr^2s^{-1} &= sr^{-1}s^{-1} = rss^{-1} = r \\ rsr^2(rs)^{-1} &= rsr^2s^{-1}r^{-1} = rsr^2sr^{-1} = rsr^2rs = rss = r \\ r^2sr^2(r^2s)^{-1} &= r^2sr^{-1}s^{-1}r^{-2} = r^2rss^{-1}r^{-2} = r^{-2} = r \\ \end{align*}

$\tau = s$

\begin{align*} ese &= s \\ rsr^{-1} &= sr^{-1}r^{-1} = sr \\ r^2sr^{-2} &= r^2sr = rs = sr^2 \\ sss^{-1} &= s \\ rss(rs)^{-1} &= r(rs)^{-1} = rsr^2 = sr \\ r^2ss(r^2s)^{-1} &= r^2sr^{-2} = sr^{-1} = sr^{2} \\ \end{align*}

$\tau = rs = sr^2$

\begin{align*} e(rs)e &= rs = sr^2 \\ r(rs)r^{-1} &= r^2sr^{-1} = s \\ r^2(rs)r^{-2} &= sr^{-2} = sr \\ s(rs)s^{-1} &= sr \\ rs(rs)(rs)^{-1} &= rs = sr^2 \\ r^2s(rs)(r^2s)^{-1} &= r^2srssr^{-2} = r^2sr^2 = s \\ \end{align*}

$\tau = r^2s = sr$

\begin{align*} e(r^2s)e &= r^2s = sr \\ r(r^2s)r^{-1} &= sr^{-1} = sr^2 \\ r^2(r^2s)r^{-2} &= rsr^{-2} = s \\ s(r^2s)s^{-1} &= sr^2 \\ rs(r^2s)(rs)^{-1} &= rsr = s \\ r^2s(r^2s)(r^2s)^{-1} &= r^2s = sr \\ \end{align*}

We see, therefore, that we have the next conjugacy classes of

D_3

\begin{equation*} \{e\}, \{r,r^2\}, \{s,sr,sr^2\} \end{equation*}