How can I show that the following pairs of permutations are in the same conjugacy class in S5:

Question

How can I show that the following pairs of permutations are in the same conjugacy class in S5?

(1,2,3,4,5) and (1,5,2,4,3)

(1)(5,3)(2,4) and (2)(1,3)(5,4)

(1,3,2) and (1,4,5)(2,3)

Answer 1

The main ideas to use are:

• that for a $k$-cycle $(c_1 \ c_2 \ \dots c _k)$, and $\sigma$ whatever permutation, you have $$\sigma (c_1 \ c_2 \ \dots c _k) \sigma^{-1} = (\sigma(c_1) \ \sigma(c_2) \ \dots \ \sigma(c_k)).$$
• $\tau \mapsto \sigma \tau \sigma ^{-1}$ is a group automorphism as being an inner automorphism.

You can use this idea to prove that: $$(2 \ 5 \ 3)(1 \ 2 \ 3 \ 4 \ 5)(2 \ 5 \ 3)^{-1} =(1 \ 5 \ 2 \ 4 \ 3)$$ $$(1 \ 2 \ 5)[(1)(5 \ 3)(2 \ 4)](1 \ 2 \ 5)^{-1}=(2)(1 \ 3)(5 \ 4)$$

The last pair in your OP cannot be in the same conjugacy class as the elements have different cycle decomposition.