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A solvable group is a group whose derived series terminates in the trivial subgroup.
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Can group solvability be detected from identities among the generators?
See Exercise 6 in Page 49 of the following book of E. I. Khukhro:
$p$-Automorphisms of Finite $p$-Groups, London Mathematical Society Lecture Note Series, 246, Cambridge University Press, Cambridge, …