Solving Algebra Problems with Symmetric Groups and Sylow Subgroups

[buzzmath]

Can anyone help me with these problems?

1. Find the number of conjugates of (1,2)(3,4) in Sn (the symmetric group of degree n) where n >= 4 and find the form of all elements commuting with (1,2)(3,4)in Sn.

2.If G is a group of order 231, prove that the 11-Sylow subgroup is in the center of G.

Thanks

 

[buzzmath]

I think I figured #2 out by saying that 1 is the only number congruent to 1 mod 11 and that divides 231 therefore the 11-sylow subgroup, N, is normal so xN=Nx for all x in G. Therefore N is in the center of G. I think this works but problem #1 is giving me a lot of trouble.