The union of two subgroups is a subgroup if and only if one is contained in the other

Let H and K be subgroups of G. Prove that H \cup K is a subgroup if and only if H \subseteq K or K \subseteq H.


The (\Leftarrow) direction is clear. To see (\Rightarrow), suppose that H \cup K is a subgroup of G and that H \not\subseteq K and K \not\subseteq H; that is, there exist x \in H with x \notin K and y \in K with y \notin H. Now we have xy \in H \cup K, so that either xy \in H or xy \in K. If xy \in H, then we have x^{-1}xy = y \in H, a contradiction. Similarly, if xy \in K, we have x \in K, a contradiction. Then it must be the case that either H \subseteq K or K \subseteq H.

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