Prove that $\mathbb{Z}_{p}$, where $p$ is a prime number, has no nontrivial subgroups

Question

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!

Answer 1

For $n$ non zero, you have $an +bp=1$. This implies that the inverse of n modulo p is a.so 1 is an element of the subgroup generated by n. Henceforth, this subgroup is $Z_p$.

Answer 2

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