- #1

Mr Davis 97

- 1,462

- 44

## Homework Statement

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

## Homework Equations

## The Attempt at a Solution

Suppose that ##H \cup K \le G##. For contradiction, suppose that neither H nor K is a subset of the other. This implies that there exists an ##h \in H## that is not in ##K## and a ##k \in K## that is not in ##H##. Since ##H## and ##K## are individually subgroups, ##h^{-1} \in H## and ##k^{-1} \in K##. Now since ##h \in H \cup K## and ##k \in H \cup K##, we know that ##hk \in H \cup K##, since we supposed it's a subgroup. We have three cases to check: If ##hk \in (H \cap K)##, then ##(hk)k^{-1} = h \in K##, a contradiction. If ##hk \in (H \cup K) - H## then ##(hk)k^{-1} = h \in K##, a contradiction. If ##hk \in (H \cup K) - K## then ##h^{-1}(hk) = k \in H##, a contradiction.