Assignment:
Qa) Let G be a group of order 203. Prove that if H is normal subgroup of order 7 in G then H<=Z(G). Deduce that G is abelian in this case.
Qb) Let P be a normal Sylow p-subgroup of G and let H be any subgroup of G. Prove that P intersect H is the unique Sylow p-subgroup of H.
Qc) Let P be in Syl_p(G) and assume N is a normal subgroup of G. Use the conjugacy part of Sylow's theorem to prove that P intesect N is a Sylow p-subgroup of N. Deduce that PN/N is a Sylow p-subgroup of G/n.
Provide complete and step by step solution for the question and show calculations and use formulas.