2024/10/30 by Ponomarenko, Ilia, Ryabov, Grigory · 1 citation
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2410.22998
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.