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Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Automorphism groups of Schur rings over cyclic groups of prime-power order Reinhard Pschel joint work with Mikhail Klin Institut fr Algebra Technische

  1. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Automorphism groups of Schur rings over cyclic groups of prime-power order Reinhard Pöschel joint work with Mikhail Klin Institut für Algebra Technische Universität Dresden Germany Ben-Gurion University of the Negev Israel Conference Modern Trends in Algebraic Graph Theory Villanova University June 2-5, 2014 Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (1/29)

  2. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Some remarks about and for My coauthor Misha Klin (unknown to him up to now) Reinhard Pöschel Institut für Algebra Technische Universität Dresden Germany Conference Modern Trends in Algebraic Graph Theory Villanova University June 4, 2014 Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (2/29)

  3. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Our roots: Issai Schur and Lev Arkad’evič Kalužnin Kiev 1971/72: Mikhail Klin — Lev Arkad’evič Kalužnin ▲❡✈ ❆r❦❛❞⑦❡✈✐q ❑❛❧✉✙♥✐♥ (31.1.1914 – 6.12.1990) Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (3/29)

  4. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Our roots: Issai Schur and Lev Arkad’evič Kalužnin Issai Schur (10.1.1875 – 10.1.1941) on the tombstone: Yeshiyahu Schur Professor of Mathematics 4 Shevet 5635 (10.1.1875) 12 Tevet 5701 (11.1.1941 [starting 10.1., 6 p.m.]) Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (3/29)

  5. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Our roots: Issai Schur and Lev Arkad’evič Kalužnin Mikhail Klin – Issai Schur – Andy Woldar Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (3/29)

  6. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Our roots: Issai Schur and Lev Arkad’evič Kalužnin Kiev 1971/72: Mikhail Klin — Lev Arkad’evič Kalužnin — Reinhard Pöschel ▲❡✈ ❆r❦❛❞⑦❡✈✐q ❑❛❧✉✙♥✐♥ (31.1.1914 – 6.12.1990) 1978 Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (3/29)

  7. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Outline Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (4/29)

  8. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Outline Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (5/29)

  9. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks S-rings S-ring (S= Schur ): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup. I. Schur used S-rings (in form of transitivity modules ) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by W. Burnside (in 1900) for a special case). I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen. S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group ) H. Wielandt 1964, Finite permutation groups. and in the papers R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring von p–Gruppen . M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings . Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

  10. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks S-rings S-ring (S= Schur ): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup. I. Schur used S-rings (in form of transitivity modules ) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by W. Burnside (in 1900) for a special case). I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen. S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group ) H. Wielandt 1964, Finite permutation groups. and in the papers R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring von p–Gruppen . M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings . Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

  11. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks S-rings S-ring (S= Schur ): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup. I. Schur used S-rings (in form of transitivity modules ) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by W. Burnside (in 1900) for a special case). I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen. S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group ) H. Wielandt 1964, Finite permutation groups. and in the papers R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring von p–Gruppen . M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings . Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

  12. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks S-rings S-ring (S= Schur ): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup. I. Schur used S-rings (in form of transitivity modules ) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by W. Burnside (in 1900) for a special case). I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen. S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group ) H. Wielandt 1964, Finite permutation groups. and in the papers R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring von p–Gruppen . M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings . Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

  13. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks S-rings S-ring (S= Schur ): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup. I. Schur used S-rings (in form of transitivity modules (“ Schurian S-rings ”)) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by W. Burnside (in 1900) for a special case). I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen. S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group ) H. Wielandt 1964, Finite permutation groups. and in the papers R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring von p–Gruppen . M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings . Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

  14. Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks S-rings S-ring (S= Schur ): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup. I. Schur used S-rings (in form of transitivity modules (“ Schurian S-rings ”)) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by W. Burnside (in 1900) for a special case). I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen. S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group ) H. Wielandt 1964, Finite permutation groups. and in the papers R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring von p–Gruppen . M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for cyclic graphs and the method of Schur rings . Villanova, June, 2014, M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

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