# How many conjugates does the permutation (123)

1. Nov 12, 2009

### dsta

1. The problem statement, all variables and given/known data
How many conjugates does the permutation (123) have in the group S3 of all permutations on 3 letters? Give brief reasons for your answers.

2. Relevant equations

3. 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."