What is an odd permutation in the centralizer of (1,2,3)(4,5,6) in $S_6$? Can you help me finding an odd permutation which commutates with $(1,2,3)(4,5,6)$ in $S_6$ ?
 A: This seems to be a good candidate: $(14)(25)(36)$
A: You have two valid answers already, but I wil try to explain how they could be derived. 
In my answer, I will let permutations act from the right.
Notice that $\sigma^{-1} (123)(456)\sigma = ( 1\sigma 2\sigma 3\sigma) (4 \sigma 5\sigma 6\sigma)$. Hence if $\sigma$ is chosen to interchange $1$ and $4$, and $2$ and $5$ and $3$ and $6,$ we will still have $\sigma^{-1} (123)(456)\sigma = (456)(123) = (123)(456).$ Hence the permutation $\sigma = (14)(25(36)$ commutes with $(123)(456),$ and is an odd permutation, as it is a product of an odd number of transpositions. 
A: $(14)(25)(36)$ commutes with $(123)(456)$ and is odd.
If $\sigma$ is a permutation of $\{1,2,4,5,6\}$, $\sigma \circ (123) \circ \sigma^{-1}=(\sigma(1) \sigma(2) \sigma(3))$.
Similarly, $\sigma \circ (123)(456) \circ \sigma^{-1}=(\sigma(1) \sigma(2) \sigma(3))(\sigma(4) \sigma(5) \sigma(6))$.
A: Saying that x and y commute is the same as saying that $yxy^{-1} = x$, that is that conjugation by y preserves x.  Conjugation by a permutation is just renaming the elements.  E.g. conjugating by $(1 2)$ just means change all 1's to 2's and vice-versa.  So you want to find a way of renaming the elements between 1 and 6 which doesn't change $(1 2 3)(4 5 6)$.  One way to do that would be to rename the elements within a give triple, e.g. conjugate by $(1 2 3)$.  But that only gives you even permutations.  The other way to do it is to interchange the roles of the two triples.  This gives you the example in the other answers.
