TITLE: Prove that $\mathbb{Z}_{p}$, where $p$ is a prime number, has no nontrivial subgroups
QUESTION [0 upvotes]: Let $p = 761$. I would like to use the fact that $761$ is a prime number to conclude that the only subgroups are $<e>$ and $\mathbb{Z}_{761}$.
I saw that many of the solutions online to this similar problem use Lagrange's theorem; however the theorem is not in my textbook, so I would prefer to use another method.
I was thinking of using the theorem stating that every subgroup of a cyclic group will be cyclic, but I am not sure how to proceed. Thank you in advance!

REPLY [1 votes]: Apply lagrange's theorem to conclude the orders of subgroups must be $1$ or $p$ and hence all subgroups must be trivial.
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