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.
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(I. N. Ponomarenko) Supported by RFBR, project No. 14-01-00376.
(G. Chen) Supported by CCNU Research Fund, project No. CCNU15A02031.
Translated from Algebra i Logika, Vol. 55, No. 1, pp. 58-74, January-February, 2016.
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Muzychuk, M.E., Ponomarenko, I.N. & Chen, G. The Schur–Wielandt Theory for Central S-Rings. Algebra Logic 55, 38–49 (2016). https://doi.org/10.1007/s10469-016-9374-9
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DOI: https://doi.org/10.1007/s10469-016-9374-9