Question as stated in the title: Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$?
If not, can you give me a counterexample?
Thanks
2026-04-06 09:25:55.1775467555
Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$?
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The answer is no. A counterexample from Derek Holt is: take $a=(1,2)$ and $b=(2,3)$ in $S_3$. The situation depends on the choice of generators as $a=(1,2)$ and $b=(1,2,3)$ in $S_3$ show.