The class-breadth conjecture revisited

Abstract

The class-breadth conjecture for groups with prime-power order was formulated by Leedham-Green, Neumann and Wiegold in 1969. We construct a new counter-example to the conjecture: it has order 219 and is a quotient of a 4-dimensional 2-uniserial space group. We translate the conjecture to p-uniserial space groups, prove that these have finite cobreadth, and provide an explicit upper bound. We develop an algorithm to decide the conjecture for p-uniserial space groups, and use this to show that all 3-uniserial space groups of dimension at most 54 satisfy the conjecture. We show that over every finite field there are Lie algebras which fail the corresponding conjecture.