please show your work and explain for this proof-type question

4. Let G be a finite group with order divisible by the prime p. Let Y denote the set of Sylow p-subgroups, with no the number of elements in Y. (a) (10 pts) Show that for g E G, then o, : Y - Y defined for P EY by o,(P) = gpg is a permutation of the set Y. (b) (10 pts) Show that 0 : G -+ Sn, defined by 0(g) = ,, where Snp is the symmetric group on no symbols, is a homomorphism. (c) (15 pts) If G is simple, prove that |G| | np! (d) (10 pts) Prove that if G is a group of order 10, then G is not simple