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Suppose G is a group, with subgroups H and K.

Prove that if H ∪ K is a subgroup of G implies that H ⊆ K or K ⊆ H.

I'm not really sure how to start this, I can prove that H ∩ K is a subgroup but I don't know how to approach the union. Any help is appreciated.

jsmith14
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1 Answers1

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The question has been changed since I wrote this -- initially there was no requirement for $H$ or $K$ to be subgroups of $G$.

Unless I've misunderstood your question, what you have written is wrong. For example, take $G = \{0,1\}$ with the operation of addition modulo 2, $H = \{0\}$ and $K = \{1\}$. Then $H \cup K = G$ which is certainly a subgroup of $G$.

Sam OT
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  • Your choices of $G$ are not groups. 0 is not invertible in either. Further, subgroups cannot have empty intersection, since they both must contain the identity of the group. – Thad Janisse Oct 12 '15 at 20:34
  • Dammit, too much stuff on rings since doing groups! Ok, my first thing still works if you change to addition modulo 2, and the second can certainly be changed slightly to work. (Note that he's changed his question since I wrote my answer -- he didn't require $H$ or $K$ to be subgroups.) – Sam OT Oct 12 '15 at 20:55
  • Made those changes. Now the answer is correct for the time it was written. – Sam OT Oct 12 '15 at 20:56
  • Unfortunately, $K$ is still not a subgroup, since $1 + 1 \notin K$. – Thad Janisse Oct 13 '15 at 03:00
  • Please re-read my answer. That will clear up any confusion. – Sam OT Oct 13 '15 at 13:03