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A New Look at the Burnside–Schur Theorem
Author(s) -
Evdokimov Sergei,
Ponomarenko Ilia
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609305004340
Subject(s) - mathematics , schur algebra , schur's lemma , combinatorics , schur's theorem , finite group , group (periodic table) , primitive permutation group , pure mathematics , discrete mathematics , symmetric group , cyclic permutation , classical orthogonal polynomials , chemistry , gegenbauer polynomials , organic chemistry , orthogonal polynomials
The famous Burnside–Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2‐transitive or isomorphic to a subgroup of a 1‐dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latter, Schur rings are introduced over a finite commutative ring, and an analogue of this statement is proved for them. Also, the finite local commutative rings are characterized in permutation group terms. 2000 Mathematics Subject Classification 20B10, 20B15, 05E99.