Group theory question

Let G be a finite group with identity e. Suppose that a an element of G has order n. Show
that H = {e, a, a^2,...,a^n-1} is a subgroup of G.

supposedly first meant to show that a^m is an element of H for
any natural number m then use Lagrange's Theorem to deduce that the order of any
element must divide the order of the group.

im really stuck with this group theory work.. if someone could run through this it would be much appreciated!

thank you

Quote:

supposedly first meant to show that a^m is an element of H for
any natural number m then use Lagrange's Theorem to deduce that the order of any
element must divide the order of the group.

Well... First of all, the subgroup generated by $a$ is $H=\{a^m:m\in \mathbb{Z} \}$ .

You can claim that $H=\{e,a,\ldots,a^{n-1}\}$ , and set out to prove this as follows:

Since $a^n=e$ , for a random integer $m$
use Euclidean division to obtain $m=kn+v$ for some integers $k,v$ with $v<n$ . Then $a^m=a^{kn+v}=a^v$ .

So, all the elements of $H$
are contained in its subset and subgroup, $S=\{e,a,\ldots,a^{n-1}\}$ . Therefore, $H=S$ .

a