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Class and Breadth of a Finite p‐Group

Class and Breadth of a Finite p‐Group CLASS AND BREADTH OF A FINITE/7-GROUP MARK CARTWRIGHT 1. Introduction Let P be a finite /?-group and let x be an element of P. If C (x) denotes the centraliser of x in P, then \P: C {x)\ is the number of distinct conjugates of x (in P). The breadth of x is then the non-negative integer b(x) such that The breadth of the group P, denoted by b(P) or just b, is the maximum of the breadths of its elements. Thus p is the size of the largest of the conjugacy classes of P. In the terminology of B. H. Neumann, p is the BFC-number of P. It was shown by P. M. Neumann, Leedham-Green and Wiegold in [4] that there is a close relationship between the breadth of a /?-group and its nilpotency class. In particular, they showed that the class c of P satisfies the inequality with equality only when b = 0, c = 1 (that is, when P is abelian). Putting this bound in a form independent of p necessitates its weakening to The class-breadth conjecture was that c^b+l, b+2 giving (for example) the dihedral group of order 2 as a natural case where http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Class and Breadth of a Finite p‐Group

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/19.5.425
Publisher site
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Abstract

CLASS AND BREADTH OF A FINITE/7-GROUP MARK CARTWRIGHT 1. Introduction Let P be a finite /?-group and let x be an element of P. If C (x) denotes the centraliser of x in P, then \P: C {x)\ is the number of distinct conjugates of x (in P). The breadth of x is then the non-negative integer b(x) such that The breadth of the group P, denoted by b(P) or just b, is the maximum of the breadths of its elements. Thus p is the size of the largest of the conjugacy classes of P. In the terminology of B. H. Neumann, p is the BFC-number of P. It was shown by P. M. Neumann, Leedham-Green and Wiegold in [4] that there is a close relationship between the breadth of a /?-group and its nilpotency class. In particular, they showed that the class c of P satisfies the inequality with equality only when b = 0, c = 1 (that is, when P is abelian). Putting this bound in a form independent of p necessitates its weakening to The class-breadth conjecture was that c^b+l, b+2 giving (for example) the dihedral group of order 2 as a natural case where

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Sep 1, 1987

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