Modern Algebra unified subgroup question

[PsychonautQQ]

Homework Statement

If H and K are subgroups of G, show HUK is a subgroup of G if and only if H < K or K < H ( the < meaning that all the elements of H are in K or all the elements of K are in H).

Homework Equations

None

The Attempt at a Solution

I believe the problem here is HUK might not be a closed group. Certainly all the elements of HUK are also in G.

If all the elements of H are in K, then hk is an element of HUK for all h,k.

If the intersection of H and K does not equal H or K, that means that hk may not be in HUK as it is not closed.

These are my thoughts so far. Am I on the right track here? How do I start turning this into a proof?

 

[Dick]

Yes, you are on the right track. Suppose h is not an element of K and k is not an element of H. Can you show hk is not an element of H or K? Use proof by contradiction.

 

[WWGD]

There is a nice, helpful result that a subset H of a group G is a subgroup of G if for any a,b in H, $ab^{-1}$ is in H.
Now if a,b are either both in A or both in B, no problem, but consider what happens when a is in H and b is in K-H ( of course then we show we must have H subset K ).

 

[PsychonautQQ]

h is not an element of K and k is not an element of H. Then hk being an element of HUK is a contradiction because.. I'm lost >.<

 

[Dick]

h is not an element of K and k is not an element of H. Then hk being an element of HUK is a contradiction because.. I'm lost >.<

Suppose hk is an element of K. Say hk=k'. Solve for h and think about it.

 

[WWGD]

Well, notice that subgroups are closed under the group operation. Let a in H, let b be in K-H and

$ab:=c \in H$ . Then $b= a^{-1} c$ , so that b is the product of elements of H, and the product of

elements in H must be in H.