= Union of two subgroups
{title2=$H\cup K\le G\ \Longleftrightarrow\ H\subseteq K\text{ or }K\subseteq H$}
The union of two <subgroups> is a <subgroup> exactly when one contains the other. Otherwise choose $h\in H\setminus K$ and $k\in K\setminus H$. Their product cannot lie in $H$, since $h^{-1}(hk)=k$ would then lie in $H$, and cannot lie in $K$, since $(hk)k^{-1}=h$ would then lie in $K$. The union therefore fails closure. Unlike the union, the <subgroup> generated by both <subgroups> always exists.
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