Permutations which make transitive groups primitive

Pedro Lopes

Open Mathematics (2009)

  • Volume: 7, Issue: 4, page 650-659
  • ISSN: 2391-5455

Abstract

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In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and certain subgroups of some affine special linear groups.

How to cite

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Pedro Lopes. "Permutations which make transitive groups primitive." Open Mathematics 7.4 (2009): 650-659. <http://eudml.org/doc/269516>.

@article{PedroLopes2009,
abstract = {In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and certain subgroups of some affine special linear groups.},
author = {Pedro Lopes},
journal = {Open Mathematics},
keywords = {Primitive groups; Permutations; Partitions; finite permutation groups; transitive permutation groups; primitive permutation groups; cycle types; generators; partitions; blocks},
language = {eng},
number = {4},
pages = {650-659},
title = {Permutations which make transitive groups primitive},
url = {http://eudml.org/doc/269516},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Pedro Lopes
TI - Permutations which make transitive groups primitive
JO - Open Mathematics
PY - 2009
VL - 7
IS - 4
SP - 650
EP - 659
AB - In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and certain subgroups of some affine special linear groups.
LA - eng
KW - Primitive groups; Permutations; Partitions; finite permutation groups; transitive permutation groups; primitive permutation groups; cycle types; generators; partitions; blocks
UR - http://eudml.org/doc/269516
ER -

References

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  7. [7] Dixon J.D., Mortimer B., Permutation groups, Graduate Texts in Mathematics, 163, Springer Verlag, 1996 Zbl0951.20001
  8. [8] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4.10; 2007 (http://www.gap-system.org) 
  9. [9] Isaacs I.M., Zieschang T., Generating symmetric groups, Am. Math. Monthly, 1995, 102(8), 734–739 http://dx.doi.org/10.2307/2974644[Crossref] Zbl0846.20004
  10. [10] Neumann P.M., Primitive permutation groups containing a cycle of prime-power length, Bull. London Math. Soc., 1975, 7, 298–299 http://dx.doi.org/10.1112/blms/7.3.298[Crossref] Zbl0319.20001
  11. [11] Wielandt H., Finite permutation groups, Academic Press, New York-London, 1964 Zbl0138.02501
  12. [12] Zieschang T., Primitive permutation groups containing a p-cycle, Arch. Math., 1995, 64, 471–474 http://dx.doi.org/10.1007/BF01195128[Crossref] Zbl0823.20001

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