Paper 3 , Section I, D

Groups | Part IA, 2021

Let GG be a finite group and denote the centre of GG by Z(G)Z(G). Prove that if the quotient group G/Z(G)G / Z(G) is cyclic then GG is abelian. Does there exist a group HH such that (i) H/Z(H)=7|H / Z(H)|=7 ? (ii) HZ(H)=6|H| Z(H) \mid=6 ?

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