A "simple" question about the order of subgroups of the group of additive integers

Question

Simple, but I cannot answer it.

Let $\mathbb Z_n$ be the additive group of integers with order $n$. Suppose $m$ is a factor of $n$. Then $\mathbb Z_n$ contains a subgroup of order $m$.

I am trying to solve this from first principles and no recourse to Lagrange or other theorems.

Answer 1

It will not get any more elementary than this: We have $\mathbb Z_n = \{0,1,2, \dotsc, n-1\}$. Show that

$$\Big\{0,\frac{n}{m}, \frac{2n}{m}, \dotsc, \frac{(m-1)n}{m}\Big\}$$

is a subgroup of $\mathbb Z_n$. It obviously has $m$ elements.