Polycyclic breadth

Origin
The notion of polycyclic breadth was introduced by Daniel Segal in his paper Polycyclic groups that appeared in the Cambridge Tracts in Mathematics, No. 82 (Cambridge University Press, 1983).

Recent work
In 2005, Tuval Foguel established some facts regarding groups of polycyclic breadth $$n$$, that generalized earlier results on supersolvable groups.

Symbol-free definition
A group is said to be of polycyclic breadth $$n$$ if it has a normal series (viz every subgroup being normal) where each of the successive quotients is Abelian with at most $$n$$ generators. The polycyclic breadth of a group is defiend as the minimum $$n$$ for which it has polycyclic breadth $$n$$.

Related subgroup properties

 * Supersolvable group: Supersolvable groups are precisely the groups of polycyclic breadth 1
 * Polycyclic group: Polycyclic groups are precisely the groups that have finite polychyclic breadth

Relation with nilpotent groups
It turns out that if $$G$$ has polycyclic breadth $$n$$, then the $$n^{th}$$ derived subgroup of $$G$$ is nilpotent. This is a (weakened) generalization of the fact that the derived subgroup of any supersolvable group is nilpotent.