Please solve this abstract algebra problem part b and c. Sylow's theorems are needed.

2. (a) Let (p : G > G' be a homomorphism, and consider a normal subgroup H' SI G'. Let H = 90'1(H') be the set of all elements of G that map to H'. Prove that H is a normal subgroup of G that contains ker go. If 90 is surjective, check that |H| = IH'| |ker90|- (b) Let G be a pgroup, that is, |G| 2 pa, for p prime and e 2 1. Prove (by induction on e) that G contains a normal subgroup of every possible order pi, for z' = 1,... ,8 1. Hint: nd a non-trivial normal subgroup of every pigroup that can start you 013\". (c) Let G be any group with order pem, where p is prime and m is not divisible by p. Prove that G contains a subgroup of order pi for each 2' = 1, . . . ,e