Elements of $\mathbb{Z}/p\mathbb{Z}$ and prove it's simple group

Question

I'm trying to understand properties of groups. Since $\mathbb{Z}/p\mathbb{Z}$ ($p$ is prime) is quotient group It should be set of all left cosets of $p\mathbb{Z}$ in $\mathbb{Z}$ which is,

$ \{ m+pn \mid m,n \in \mathbb{Z} \} $

Then I set an $H\subseteq p\mathbb{Z}$.Then it must satisfy,

$ \forall h_1,h_2\in H:h_1h_2^{-1} \in H$

I'm stuck here. I don't what to do. Please help me.

Answer 1

The elements are each of the form

$$[a]_p:=\{b\in\Bbb Z: p\mid a-b\}$$

for $a\in \Bbb Z$; so: $$[0]_p, [1]_p, \dots, [p-1]_p.$$

It's simple due to Lagrange's Theorem: since $p$ is prime, the only subgroups of $\Bbb Z/p\Bbb Z$ are the trivial ones.