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