Exercise 4.3.11 (Find $\tau$ such that $\sigma_2 =\tau \sigma_1 \tau^{-1}$)

Proof. We determine whether ${\sigma }_1$ and ${\sigma }_2$ are conjugate in the following cases:

(a)

${\sigma }_1=(12)(345)$ and ${\sigma }_2=(123)(45)=(45)(123)$.

Put $\tau =\left(\begin{array}{ccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \end{array}\right)=(14253)$. Then by Proposition 10

$$\tau {\sigma }_1{\tau }^{-1}=(45)(123)={\sigma }_2,$$

so ${\sigma }_1$ and ${\sigma }_2$ are conjugate.

(b)

${\sigma }_1=(15)(372)(106811)$ and ${\sigma }_2=(37510)(49)(13112)$. Then $$\begin{array}{llll}\hfill {\sigma }_1& =(4)(9)(12)(13)(15)(372)(106811)& \hfill & \\ \hfill {\sigma }_2& =(1)(6)(8)(12)(49)(13112)(37510)& \hfill & \end{array}$$

Put $\tau =\left(\begin{array}{ccccccccccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill & \hfill 10\hfill & \hfill 11\hfill & \hfill 12\hfill & \hfill 13\hfill \\ \hfill 4\hfill & \hfill 2\hfill & \hfill 13\hfill & \hfill 1\hfill & \hfill 9\hfill & \hfill 7\hfill & \hfill 11\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 3\hfill & \hfill 10\hfill & \hfill 8\hfill & \hfill 12\hfill \end{array}\right)=(14)(31312859671110)$.

Then

$$\tau {\sigma }_1{\tau }^{-1}=(49)(13112)(37510)={\sigma }_2,$$

so ${\sigma }_1$ and ${\sigma }_2$ are conjugate.

(c)

${\sigma }_1=(15)(372)$ and ${\sigma }_2={\sigma }_1^3$.

Then ${\sigma }_2=(15)$ and ${\sigma }_1$ have not the same cycle type, so ${\sigma }_1$ and ${\sigma }_2$ are not conjugate.

(d)

${\sigma }_1=(13)(246)$ and ${\sigma }_2=(35)(24)(56)$.

Note that

$${\sigma }_2=(35)(24)(56)=(24)(35)(56)=(24)(356)$$

has the same cycle type than ${\sigma }_1$.

Put $\tau =\left(\begin{array}{cccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 6\hfill \end{array}\right)=(12345).$

$$\tau {\sigma }_1{\tau }^{-1}=(24)(356)={\sigma }_2,$$

so ${\sigma }_1$ and ${\sigma }_2$ are conjugate.