Expert: Paul Klarreich - 11/8/2008

Question
?Let H and K be subgroups of a group G where the order of H=9 and the order of K=12 and where the index [G:H intersect K] is not equal to the order of G. Find the order of H intersect K.

Well I know that H is a subgroup of G and K is a subgroup of G. Using the index and Lagrange's theorem (the order of H divides the order of G). This is where I am stuck. Do I use the order of H intersects K divides the order of H=9 and do the same for the order of H intersects K divides the order of K=12? I can't figure out the answer, but I think it is 3. Your help would be greatly appreciated.

Answer
Questioner:   Jacob
Category:  Advanced Math
Private:  No
 
Subject:  Abstract Algebra
Question:  Let H and K be subgroups of a group G where the order of H=9 and the order of K=12 and where the index [G:H intersect K] is not equal to the order of G. Find the order of H intersect K.

Well I know that H is a subgroup of G and K is a subgroup of G. Using the index and Lagrange's theorem (the order of H divides the order of G). This is where I am stuck. Do I use the order of H intersects K divides the order of H=9 and do the same for the order of H intersects K divides the order of K=12? I can't figure out the answer, but I think it is 3. Your help would be greatly appreciated.
............................
Hi, Jacob,

I think you are right.  I am somewhat uncomfortable with this, but it seems to me that:

HK [intersection] can easily be shown to be a group.
..............
Is it easy?

e is obviously in both H and K.
x in H --> x* [inverse] in H, and
x in K --> x* in K.
So  x in HK --> x* in HK.

Maybe.
................

Now would order(HK) have to divide order(H) and order(K)?  So either order(HK) = 1 or order(HK) = 3.

Now you could use that index business.