The Schur-Wielandt theory for central S-rings

TL;DR

This paper extends key theorems from abelian group S-rings to central S-rings in non-abelian groups, broadening understanding of their structure and properties.

Contribution

It proves the Schur and Wielandt theorems for central S-rings and generalizes the concept of B-groups, linking them to Camina groups.

Findings

Schur theorem on multipliers holds for central S-rings.

Wielandt theorem on primitive S-rings extends to central S-rings.

Camina groups are generalized B-groups, unlike most simple groups.

Abstract

Two basic results on the S-rings over an abelian group are the Schur theorem on multipliers and the Wielandt theorem on primitive S-rings over groups with a cyclic Sylow subgroup. None of these theorems is directly generalized to the non-abelian case. Nevertheless, we prove that they are true for the central S-rings, i.e., for those which are contained in the center of the group ring of the underlying group (such S-rings naturally arise in the supercharacter theory). We also generalize the concept of a B-group introduced by Wielandt, and show that any Camina group is a generalized B-group whereas with few exceptions, no simple group is of this type.