Polycyclic breadth

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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 n, that generalized earlier results on supersolvable groups.

Definition

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.

Definition with symbols

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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 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.

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)