stabilizers and orbits of elements in $S_4$

Question

Let $S_4$ be the group of symmetries of $4$ letters, and let $S_4$ act on itself by conjugation. Let $\sigma = (12)$, $\tau = (123)$, and $\rho = (1234)$.

1. Find the stabilizers of the elements $\sigma$, $\tau$, and $\rho$.
2. Find the orbits of the elements $\sigma$, $\tau$, and $\rho$.

Answer 1

1.

The stabilizer of $\sigma$ consists of the permutations that commute with it. Hence $$\text{Stab}(\sigma) = \{\text{Id},(34),(12),(12)(34)\}.$$Similarly,$$\text{Stab}(\tau) = \{\text{Id}, (123),(132)\},$$$$\text{Stab}(\rho) = \{\text{Id},(1234),(1432),(13)(24)\}.$$

2.

Conjugacy classes in $S_n$ correspond to cycle types. Thus,$$\text{orbit of }\sigma=\text{set of transpositions,}$$$$\text{orbit of }\tau=\text{set of }3\text{-cycles,}$$$$\text{orbit of }\rho=\text{set of }4\text{-cycles.}$$