# Finite cyclic and homomorph-containing implies order-unique

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## Statement

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

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

## Proof

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