Question #203250

Prove that a cyclic group with only one generator can have at most 2 elements.


1
Expert's answer
2021-06-08T09:00:34-0400

Let us prove that a cyclic group GG with only one generator can have at most 2 elements. If GG is infinite cyclic group, then GG is isomorphic to the group (Z,+)=1=1(\mathbb Z,+)=\langle1\rangle=\langle -1\rangle, and hence it has two generators. Therefore, GG must be a finite cyclic group. Let aa be a generator of a cyclic group , and let a=n.|a|=n. Taking into account that a1=n|a^{-1}|=n and aa is a unique generator of G,G, we conclude that a1=a.a^{-1}=a. It follows that a2=e,a^2=e, and hence GG is a cyclic group of order 2, or GG is a trivial group. We conclude that GG can have at most 2 elements.


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