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Transitive Permutation Groups Without Semiregular Subgroups
Author(s) -
Cameron Peter J.,
Giudici Michael,
Jones Gareth A.,
Kantor William M.,
Klin Mikhail H.,
Marušič Dragan,
Nowitz Lewis A.
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003484
Subject(s) - permutation group , conjecture , permutation (music) , combinatorics , group (periodic table) , mathematics , primitive permutation group , transitive relation , affine transformation , class (philosophy) , product (mathematics) , counterexample , cyclic permutation , discrete mathematics , symmetric group , computer science , pure mathematics , physics , artificial intelligence , geometry , quantum mechanics , acoustics
A transitive finite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and affine group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups. Part of the motivation for studying this class of groups was a conjecture due to Marušič, Jordan and Klin asserting that there is no elusive 2‐closed permutation group. It is shown that the constructions given will not build counterexamples to this conjecture.