Infinite Order of Non-Identity Elements

[SpringPhysics]

Homework Statement

G is an abelian group with H the subgroup of elements of G with finite order. Prove that every non-identity element in G/H has infinite order.

Homework Equations

The Attempt at a Solution

Suppose gH in G/H has order n.
Then (gH)ⁿ = gⁿH so gⁿ is in H.
Then there is some m > 0 such that g^nm = e.
So g is in H.

At this point, I am stuck - I do not know how to show that g must be the identity element.

EDIT: Figured it out.

 

[lippyka]

Since g is in H, then g has a finite order m. Since (gH)n = eH = H, then m must divide n.But this is a contradiction since the order of gH is n. Therefore g must be the identity element and the statement is proven.