Prove the order of an element divides the order of the group using cosets I know how to use cyclic groups and Lagrange's theorem to prove this, but I don't know how to use the notion of cosets to prove this.
 A: To try to clean up some of the comments, I'll make an attempt at an answer. I'm not sure if this qualifies as "using cosets".
The actual proof is going to depend on the statement of Lagrange's theorem you have; my preferred version (for finite groups) is the following:

If $G$ is a finite group, $H$ is a subgroup, and $|G:H|$ is the number
  of (left, for sake of argument) cosets of $H$, then
$$|G|=|H||G:H|$$

Thus, given $a\in G$, to prove that $|a|$ divides $|G|$ it is enough to find a subgroup $H$ such that $|H|=|a|$. But as you mention in the comments, $\langle a\rangle$ is such a subgroup, so you have the result.
If your version of Lagrange's theorem says something else, you may need to do more work; for example, it might simply state that $|G:H|$ divides $|G|$. In this case you will need to look at a proof, and see that (as is very likely!) it actually proves the identity I gave above. (Perhaps this is what you mean by using cosets; you can show that the number of cosets of $\langle a\rangle$ in $G$ divides $|G|$, and the result of this division is $|\langle a\rangle|=|a|$, so $|a|$ divides $|G|$ too.)
