Problem:Prove or provide a counter example.

If H and K are subgroups of a group G, then $\displaystyle H \cup K$ is a subgroup of G.

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I had the set of generators 0 through 17 for $\displaystyle \mathbb{Z}_{18}$ sitting in front of me from a previous problem.

I saw that

$\displaystyle <2> = \left\{0,2,4,6,8,10,12,14,16\right\}$,

$\displaystyle <9> = \left\{0,9\right\}$.

So I said $\displaystyle <2>=H$ and $\displaystyle <9>=K$.

Therefore, $\displaystyle H \cup K$ would be

$\displaystyle \left\{0,2,4,6,8,9,10,12,14,16\right\}$

But this isn't closed under the group operation. $\displaystyle 9+2 \hspace{1mm} mod \hspace{1mm} 18=11$ and $\displaystyle 11 \notin H\cupK$

So am I correct in saying that $\displaystyle H \cup K$ will not always be a subgroup of G?

Even if I'm correct, I'm a little concerned. If I had not had the generator list in front of me, I wouldn't have immediately spotted the counterexample. Is there any mathematically concrete way to go about this?