Answer to Question #188488 in Algebra for Priyanka

Question #188488

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


1
Expert's answer
2021-05-07T10:40:41-0400

Let us prove that a cyclic group with only one generator can have at most 2 elements. Let "a" be a generator of a cyclic group. Since the order of "a^{-1}"is equal to the order of "a," we conclude that "a^{-1}"is also a generator, and hence "a^{-1}=a." It follows that "a^2=e" after multiplying both parts by "a." Consequently, a cyclic group can have at most 2 elements.


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