Let $G=\Bbb Z$ and $H =m\Bbb Z$. What is $(G : H)$?

Question

Let $G=\Bbb Z$ and $H =m\Bbb Z$. What is $(G : H)$? I think the answer is $m$, where $m$ is a positive integer. Because $-m\Bbb Z$ and $m\Bbb Z$ are identical. Help me

Answer 1

$\mathbb Z/m \mathbb Z$ is an $m$-element group (the integers mod m) denoted $\mathbb Z_m$...

Answer 2

Hint: $$(G:H)= \bigl|\mkern1mu G/H\mkern 1mu\bigr|.$$