Consider the group $s_3$ of example 8.7

Question

Find the cyclic subgroups $<\rho_1>, <\rho_2>, and <\mu_1>$ of $S_3$. Elements of $S_3$.

I know the answer is suppose to be $<\rho_1> = <\rho_2> = \{\rho_0, \rho_1, \rho_2 \}$ and $<\mu_1> = \{ \rho_0, \rho_1 \}$. I'm not sure if my work shows this. For $ <\rho_1, <\rho_2>$ is it sufficient to say that

$(231)(123)= (231)$

$(231)(231)= (312)$ & $(312)(231)=(123)$ & $(123)(231)=(231)$

We can say $<\rho_1> = <\rho_2> = \{\rho_0, \rho_1, \rho_2 \}$

Answer 1

There are exactly four cyclic subgroups: $\{\rho_0,\rho_1, \rho_2\}$ (of order $3$), and $\{\rho_0,\mu_1\}$, $\{\rho_0,\mu_2\}$, $\{\rho_0,\mu_3\}$ each of order $2$.

Answer 2

Note that $\mu_1 = (1 2)$, so $\mu_1 \mu_1 = (1 2)(1 2) = (1) = \rho_0$, which is the identity element for $S_3$.

So $\langle \mu_1 \rangle = \{ \rho_0, \mu_1 \}$.