# Group Theory

1. Apr 11, 2009

### 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?