How many conjugates does the permutation (123)

[dsta]

Homework Statement

How many conjugates does the permutation (123) have in the group S₃ of all permutations on 3 letters? Give brief reasons for your answers.

Homework Equations

The Attempt at a Solution

Answers were provided for this question, but after going carefully through it, I still don't know what's going on at one point in the answer, which may prove crucial to my understanding of the problem (where I've put question marks). Below is the solution provided:

"We can use the stabilizer-orbit relationship to see that the size of the conjugacy class of (123) is equal to the index of the stabilizer. But the stabilizer is a subgroup of S₃ which contains at least <(123)> [? why]. If it were to contain more then it must be all of S₃ (by Lagrange's theorem) and so contain, for example, (12). But (12)(123)(12)^-1 = (132) and so the centralizer is exactly <(123)> [?]. Since it has order 3, it also has index 2 and so (123) has 2 conjugates."

 

[kurt.physics]

I'm not sure why the stabilizer contains (123) and why the centralizer is exactly <(123)>. Can someone explain this? Thanks a lot.