Trouble understanding Cosets and Lagrange's Theorem Let $G$ be a group and $H$ a subgroup of $G$
I think all the elements of $h \in H$ multiplied (on the left or right) by one element $g\in G$ forms a coset.
Intuitively I can see that $|H|$ = $|gH|$=$|Hg|$ -multiplying all the elements of $H$ does not change the number of elements
So the size of each coset depends on the size of $H$ you chose. And yet it turns out that the the size of $H$ always divides the size of $G$ (Lagrange's Theorem) What if you choose $H$ such that its order will not divide the order of $G$?
 A: What you're doing here is outlining the standard proof that |H| divides |G| for finite groups.  (This is Lagrange's theorem, I think.)
Yes, your notion of a coset is correct, and all cosets of H have the same number of elements.  Add to that that every element of G is in a coset, and that two cosets are either identical or have no elements in common, then you've got all you need to complete the proof.
The only cases where |H| cannot divide |G| is when G is an infinite group.  :-)
A: It sounds like you're forgetting that $H$ has to be a subgroup, not just a subset. It is true that if $H$ is any subset, then $|gH| = |H|$ for all $g \in G$. But if $H$ is not a subgroup, then $gH$ is just another subset (not a coset), and the argument behind Lagrange's theorem does not work.
The proof of Lagrange's theorem begins: if $H$ is a subgroup, then each coset $gH$ has the same size as $H$, and furthermore, two cosets which are not the same, must have no elements in common. This last statement is not necessarily true if $H$ is not a subgroup, but it is very important, since it lets us write
$$|G| = |g_1H| + |g_2H| + \ldots + |g_rH| = r|H|.$$
for some cosets $g_1H$, etc, which is how we deduce that $|G|/|H| = r$ is an integer.
An example of where this goes wrong: Let $G = \{e,x,x^2\}$ where $x^3 = e$, and let $H = \{e,x\}$, which is not a subgroup. Then the subset $x^2H = \{x^{2},e\}$ has one element in common with $H$. So $G = H \cup x^{2}H$ but $|G| \neq |H| + |x^{2}H|$.
