# Infinite Order of Non-Identity Elements

#### 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:
Related Calculus and Beyond Homework News on Phys.org

### Want to reply to this thread?

"Infinite Order of Non-Identity Elements"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving