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## A conjecture on solvablity of finite groups

Suppose $G$ is a finite group and $A$ an abelian subgroup. Suppose for some natural number $n\geq 2$, elements of $\gamma_n(G)$ have the form $[a, x]$ where $a\in A$ and $x\in G$. Then $G$ is solvable.

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 A similar statement is true for finite dimensional Lie algebras of characteristic zero. See my short note in Bull. Australian Math. Soc.(2011): A note on derivations of Lie algebras. – M Shahryari Dec 5 at 17:03 and what is $\gamma_n(G)$? – Dima Pasechnik Dec 5 at 17:38 @Dima: it is the $n$-th term of the lower central series. – M Shahryari Dec 5 at 18:34

Let $H$ be a finite group, $a$ a fixed point free automorphism of $H$, and $A = \langle a \rangle$. Let $G$ be the semidirect product of $H$ by $A$. Then it is well known that every element of $H$ is of the form $[a, h]$, for $h \in H$, so that $H = \gamma_2(G)$.
Now it is indeed true that $H$ (and thus $G$) is soluble in this case, but the proof requires the classification of finite simple groups.