Let G be any group with no proper , nontrivial subgroups and assume abs value(G)>1. Prove that G must be isomorphic to Z_p for some prime p.

I know we have an isomorphism if a group is 1-1, onto, and the homomorphism property holds.

G is cyclic means G=<a>, where a is a generator.

I know a cyclic group of order n has exactly one subgroup of order m for each positive divisor m of n.