# Infinite Order of Non-Identity Elements

1. Feb 14, 2012

### SpringPhysics

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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 gnm = 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.

Last edited: Feb 14, 2012