Assignment:
Verify that if H is a subgroup of G, and a ? G, then aHa^-1 is a subgroup of G.
Prove that if H is a finite subgroup of G, and a ? G, then |aHa^-1| = |H|. (Suggestion: The mapping h ? aha^-1 is one-to-one.)
Explain why if H is a Sylow p-subgroup of a finite group, then so is each conjugate of H.
Prove that if a finite group has only one Sylow p-subgroup for some prime p, then that subgroup must be normal.
Provide complete and step by step solution for the question and show calculations and use formulas.