In the solution to this problem I've been given by my professor, he says that $g^2$ generates G if n is odd but not if n is even. I am confused about this since I came up with the example that $Z/8$ is generated by 3 and also by 9, since 9 is congruent to 1, but this contradicts his answer. Any help/guidance would be much appreciated.
2026-03-28 01:48:51.1774662531
If G is a finite cyclic group of size n is generated by g, is it generated by $g^2$?
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You are confused because $g^2$ doesn't mean "$g$ to the power of $2$", the concept of "power" isn't even well defined for an arbitrary group. Instead, it is literally defined as $gg$. Now in $\mathbb{Z}/8\mathbb{Z}$ addition is the group operation. And so it becomes $g+g$. With that you have $3+3=6$ and $6$ is of order $4$, thus it doesn't generate $\mathbb{Z}/8\mathbb{Z}$.