Finding Centraliser of (12)(34) in S_4

[latentcorpse]

What is the centraliser of (12)(34) in $S_4$. check your answer is consistent with the size of the conjugacy class.

so i found the conjugacy class had size 3.

and hence by orbit stabiliser the centraliser must have size 8.

now under conjugacy action $C(g)=\{g \in G | gh=hg\} = Stab(g)$

clearly (12)(34) is comjugate to the identity and the 2 other elements with cycle type (2,2) so that's me found 4 elements. according to the answers the other elements in the centraliser are two 4 cycles and 2 transpositions - but how can we tell which 4 cycles and which transpositions these are?

also, since elements of cycle type (2,2) are conjugate to (12)(34) and so these will be in the conjugacy class but according to the above they are also in the stabiliser/centraliser - why is this?

 

[lippyka]

The centraliser of (12)(34) in S_4 is the set of all permutations that commute with (12)(34), i.e. those that leave it invariant under conjugation. These are: (12)(34), (13)(24), (14)(23), (12)(43), (13)(42), (14)(32), (23)(41), (24)(31). The size of the conjugacy class is 3, since (12)(34) is conjugate to itself and the other two elements of cycle type (2,2), which are (13)(24) and (14)(23). The stabiliser or centraliser of (12)(34) then has size 8, since by the Orbit-Stabiliser Theorem, the size of the centraliser is equal to the size of the conjugacy class multiplied by the size of the stabiliser. The four 4-cycles and two transpositions can be determined by noting that the 4-cycles are conjugates of (12)(34), and the transpositions are conjugates of the pairs of transpositions that form a 2-cycle in (12)(34).