Let $F_4$ be the free group on the letters $a,b,c,d$. I would like to prove that the element $[a,b][c,d] = aba^{-1}b^{-1}cdc^{-1}d^{-1}$ is not equal to $[x,y] = xyx^{-1}y^{-1}$ for any elements $x,y \in F_4$.
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1Note that it is sufficient to find any group in which the set of commutators is not a subgroup (since you can construct a homomorphism from $F_4$ to that group sending $[a,b]$ and $[c,d]$ to two commutators whose product is not a commutator). – stewbasic Mar 27 '18 at 05:27
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1Also it seems this has been asked several times; probably the most useful instance is https://math.stackexchange.com/q/59816/197161 – stewbasic Mar 27 '18 at 05:38
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@stewbasic Ahh, right - thanks!! – user101010 Mar 27 '18 at 05:51