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- Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.Let H be a normal cyclic subgroup of a finite group G. Prove that every subgroup K of H is normal in G.Let H be a subgroup of the group G. Prove that the index of H in G is the number of distinct right cosets of H in G.