Prove that a cyclic group with only one generator can have at most 2 elements.
Let us prove that a cyclic group with only one generator can have at most 2 elements. If is infinite cyclic group, then is isomorphic to the group , and hence it has two generators. Therefore, must be a finite cyclic group. Let be a generator of a cyclic group , and let Taking into account that and is a unique generator of we conclude that It follows that and hence is a cyclic group of order 2, or is a trivial group. We conclude that can have at most 2 elements.
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