On Wielandt's Join Theorem for fusion systems and localities

TL;DR

This paper extends Wielandt's classical subgroup theorem to the contexts of fusion systems and regular localities, providing new theoretical insights and group-theoretical results.

Contribution

It proves versions of Wielandt's theorem for fusion systems and localities, bridging classical group theory with these modern algebraic structures.

Findings

Wielandt's theorem is valid for regular localities.

Wielandt's theorem is valid for fusion systems.

A new group-theoretical result of independent interest is proved.

Abstract

Saturated fusion systems are categories generalizing important aspects of conjugacy of $p$-subgroups in finite groups. It was shown by Chermak that there are group-like structures called regular localities associated to saturated fusion systems. Both the theory of fusion systems and the theory of regular localities are developed in analogy to the theory of finite groups. In this paper we focus on a classical theorem of Wielandt, which states that any two subnormal subgroups of a finite group $G$ generate a subnormal subgroup of $G$. We prove versions of this theorem for regular localities and for fusion systems. Along the way we prove also a purely group-theoretical result which may be of independent interest.