Finite cyclic and homomorph-containing implies order-unique

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Statement

Suppose G is a group and H is a finite cyclic subgroup of G that is also a Homomorph-containing subgroup (?). Then, H is an Order-unique subgroup (?) of G: there is no other subgroup of G isomorphic to H.

When G is a finite group, this is equivalent to saying that H is an Index-unique subgroup (?) of G.

Proof

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