Can a Group Be the Union of Two Proper Subgroups?

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SUMMARY

The discussion centers on the mathematical proof that the union of two proper subgroups, H and K, of a group G cannot equal G itself. It is established that if either H is a subset of K or K is a subset of H, then the union is trivially equal to G. However, if neither inclusion holds, the proof requires examining elements from both subgroups and their products. This leads to the conclusion that H union K cannot encompass all elements of G.

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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
 
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spotsymaj said:
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?
 

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