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**Soltras** All right here's my latest one that's already consumed a healthy stack of papers trying to work out. This one is on p.106 of Hungerford's Algebra text, #3(a).

If H and K are subgroups of G let (H,K) be the subgroup of G generated by the elements { $\displaystyle hkh^{-1}k^{-1} : h \in H, k \in K$ }. **Show that (H,K) is normal in H v K. **

(H v K is the *join *of H and K, the subgroup generated by elements in the union of H and K).

Here's the progress I've made. First of all, obviously $\displaystyle (H,K) \subset H \vee K$. In fact, it's a subset of the commutator subgroup of $\displaystyle H \vee K$. If I can show containment the other way (which I'm not even sure is correct) then normal property follows because commutator subgroups are normal.

Instead, I've tried letting $\displaystyle x \in H \vee K$ and looking at $\displaystyle x(H,K)x^{-1}$. It should be enough to show that $\displaystyle (xhx^{-1})(xkx^{-1})(xh^{-1}x^{-1})(xk^{-1}x^{-1}) \in (H,K)$, which would be easy if N and K are normal in $\displaystyle N \vee K$, but this isn't given, so it's been difficult. Surely I'm just overlooking some proposition, which is why I'm seeking help here from others who may be more comfortable with group theory. Any help would be appreciated.