# Is there a commutator-theoretic criterion for supersolvability of a group?

A group $G$ is nilpotent if and only if there is a $c\gt 0$ such that the $(c+1)$st term of the lower central series is trivial. A group $G$ is solvable if and only if there is a $c\gt 0$ such that the $c$th term of the derived series is trivial.

Is there some similar criterion for supersolvability, or at least one which is purely commutator-theoretic?

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## 1 Answer

Nilpotent groups of a given class and solvable groups of a given class form varieties, each of these varieties is defined by commutator identities. Supersolvable groups of any class do not form a variety of groups. Moreover it is not a union of varieties because there exists a supersolvable group which generates a variety where not all groups are supersolvable. So there is no definition in terms of identities - commutator or not.

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Thanks! A colleague asked me, and I thought as much but wasn't sure. I appreciate it. –  Arturo Magidin Oct 11 '10 at 3:09