Formations of finite groups with the M. Hall property

TL;DR

This paper constructs the first examples of certain group formations with specific properties, highlighting distinctions between being arboreous, Hall, freely indexed, and locally extensible, and discusses open questions in the area.

Contribution

It introduces the first known formations that are arboreous and Hall but not freely indexed, and also constructs solvable formations with these properties that are not locally extensible.

Findings

First examples of arboreous but not freely indexed formations.

First solvable formations that are freely indexed and arboreous but not locally extensible.

Discussion of open questions in formation theory.

Abstract

The first examples of formations which are arboreous (and therefore Hall) but not freely indexed (and therefore not locally extensible) are found. Likewise, the first examples of solvable formations which are freely indexed and arboreous (and therefore Hall) but not locally extensible are constructed. Some open questions are also mentioned.