Notes on $B$-groups

TL;DR

This paper explores the distinction between B-groups and BS-groups, providing examples of B-groups that are not BS-groups, thereby advancing understanding of their structural properties.

Contribution

It introduces the concept of BS-groups and demonstrates infinitely many B-groups that do not qualify as BS-groups, expanding the classification of these groups.

Findings

Existence of infinitely many B-groups that are not BS-groups.

Analysis of Schur rings reveals stronger properties for certain groups.

Provides new examples differentiating B-groups from BS-groups.

Abstract

Following Wielandt, a finite group $G$ is called a $B$-group (Burnside group) if every primitive group containing a regular subgroup isomorphic to $G$ is doubly transitive. Using a method of Schur rings, Wielandt proved that every abelian group of composite order which has at least one cyclic Sylow subgroup is a $B$-group. Since then, other infinite families of $B$-groups were found by the same method. A simple analysis of the proofs of these results shows that in all of them a stronger statement was proved for the group $G$ under consideration: every primitive Schur ring over $G$ is trivial. A finite group $G$ possessing the latter property, we call $BS$-group (Burnside-Schur group). In the present note, we give infinitely many examples of $B$-groups which are not $BS$-groups.