91Ó°ÊÓ

Let \(G\) be a finite \(p\) -group, and let \(H\) be a nontrivial normal subgroup of \(G\). Prove that \(H \cap Z(G) \neq 1\).

Prove that a Sylow 2-subgroup of \(A_{5}\) has exactly 5 conjugates.

Exhibit all the subgroups of \(S_{4} ;\) aside from \(S_{4}\) and 1 , there are 26 of them.

Prove that any simple group \(G\) of order 60 is isomorphic to \(A_{5}\). (Hint. If \(P\) and \(Q\) are distinct Sylow 2-subgroups having a nontrivial element \(x\) in their intersection, then \(C_{G}(x)\) has index 5 ; otherwise, every two Sylow 2 -subgroups intersect trivially and \(N_{G}(P)\) has index \(5 .\) )