Be clear on your proofs. Give all details and explain why each claim you make is justified. Write in complete sentences. Give the level of detail in your solutions which I have given in the solutions I have posted. Do not work with anyone else; if you have specific questions then e-mail me. 5. (Exercise II.5.5) If P is a normal Sylow p-subgroup of a finite group G and f: G G is an endomorphism (that is, a homomorphism mapping G to G), then f(P) < P. HINT: Use the First Sylow Theorem (Theorem II.5.7) to show that f(P) is a subgroup of some Sylow p-subgroup. Use the Second Sylow Theorem (Theorem II.5.10) to show that P is the only Sylow p-subgroup of G.