Paper 3, Section II, E

Groups | Part IA, 2017

Let NN be a normal subgroup of a finite group GG of prime index p=G:Np=|G: N|.

By considering a suitable homomorphism, show that if HH is a subgroup of GG that is not contained in NN, then HNH \cap N is a normal subgroup of HH of index pp.

Let CC be a conjugacy class of GG that is contained in NN. Prove that CC is either a conjugacy class in NN or is the disjoint union of pp conjugacy classes in NN.

[You may use standard theorems without proof.]

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