Polycyclic breadth
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
This article defines an arithmetic function on a restricted class of groups, namely: polycyclic groups
History
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 , that generalized earlier results on supersolvable groups.
Definition
Symbol-free definition
A group is said to be of polycyclic breadth if it has a normal series (viz every subgroup being normal) where each of the successive quotients is Abelian with at most generators. The polycyclic breadth of a group is defiend as the minimum for which it has polycyclic breadth .
Definition with symbols
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]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
Facts
Relation with nilpotent groups
It turns out that if has polycyclic breadth , then the derived subgroup of is nilpotent. This is a (weakened) generalization of the fact that the derived subgroup of any supersolvable group is nilpotent.
References
- A Generalization of Supersolvability by Tuval Foguel, Glasgow Mathematical Journal, Vol 47, 2005, Pages 249-253
- Polycyclic groups by Daniel Segal, Cambridge Tracts in Mathematics, No. 82, Cambridge University Press (1983)