A conjecture on B-groups

TL;DR

This paper proposes a conjecture linking the nilpotency of a finite group to its largest quotient B-group and provides a proof under the solvability condition, connecting it to biset-functors.

Contribution

It introduces a new conjecture relating group nilpotency to B-groups and proves it for solvable groups, linking it to biset-functor theory.

Findings

Conjecture holds for solvable groups.

Nilpotency of G is equivalent to nilpotency of β(G).

Kernel of restrictions to nilpotent subgroups forms a biset-subfunctor.

Abstract

In this note, I propose the following conjecture: a finite group G is nilpotent if and only if its largest quotient B-group \beta(G) is nilpotent. I give a proof of this conjecture under the additional assumption that G be solvable. I also show that this conjecture is equivalent to the following: the kernel of restrictions to nilpotent subgroups is a biset-subfunctor of the Burnside functor.