Prove that if and are normal subsets of a Sylow -subgroup of a finite group then is -conjugate to if and only if is -conjugate to . Deduce that two elements in the center of are conjugate in if and only if they are conjugate in . (A subset is normal in if .)
Let with and .
The direction is immediate. To see , suppose is -conjugate to ; then there exists such that and . Note that . Now . (@@@)
Now suppose . In this case, and , so that and are normal in . By the above argument we have that and are -conjugate if and only if they are -conjugate.