The Schur–Wielandt theory for central $S$-rings

Two basic results on $S$-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. Neither of these is directly generalized to the non-Abelian case. Nevertheless, we prove that the two theorems are true for central $S$-rings over any group, i.e., for $S$-rings that are contained in the center of the group ring of that group (such $S$-rings arise naturally in the supercharacter theory). Extending the concept of a $B$-group introduced by Wielandt, we show that every Camina group is a generalized $B$-group, whereas simple groups, with few exceptions, cannot be of this type.