• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Order of subgroup of an abelian group

  • Thread starter H12504106
  • Start date
6
0
1. Homework Statement

Suppose H and K are subgroups of an abelian group G (not neccessarily finite). Let the order of H and K be a and b respectively. Prove that there exists a subgroup of order L, where L = lcm(a,b).

2. Homework Equations

Product Formula: |HK|/|H| = |K|/|H intersect K|
lcm(a,b)*gcd(a,b)=ab
Lagrange theorem
A finite abelian group G has a subgroup of order d for every divisor d of |G|


3. The Attempt at a Solution

Using the product formula, i obtain: |HK||H intersect K| = ab
Then using largrange theorem, i have |H intersect K| divides |H| and |H intersect K| divides |K|. Hence, |H intersect K| divides ab. However, i am unable to proceed further. Or is there another method to solve this problem

Thank You.
 

Want to reply to this thread?

"Order of subgroup of an abelian group" You must log in or register to reply here.

Related Threads for: Order of subgroup of an abelian group

  • Posted
Replies
7
Views
662
Replies
1
Views
7K
Replies
5
Views
2K
Replies
4
Views
1K
Replies
4
Views
5K
  • Posted
Replies
3
Views
6K
  • Posted
Replies
1
Views
3K
Replies
3
Views
313

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
Top