Can a Group Be the Union of Two Proper Subgroups?

[spotsymaj]

The following problem was given on a test of mine and I got it completely wrong. If anyone can help me with solving this problem that would be great

Let H and K be a subgroup of G, such that H is not equal to G and K is not equal to G . Prove that H union K is not equal to G. Hint: A group cannot be the union of two proper subgroups

 

[jgens]

Let H and K be a subgroup of G, such that H is not equal to G and K is not equal to G . Prove that H union K is not equal to G. Hint: A group cannot be the union of two proper subgroups

If either H ≤ K or K ≤ H, then the desired result holds trivially. Now suppose that neither of these inclusions hold. Fix an element h in H not contained in K and another element k in K not contained in H. Is the product hk in either H or K?