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Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside

Part of: Representation theory of groups Permutation groups

Published online by Cambridge University Press:  27 May 2019

MARK WILDON
MARK WILDON*
Affiliation:
Dept. of Mathematics, Royal Holloway, University of London, Egham Hill, Egham Tw20 OEX, U.K.
*
e-mail: mark.wildon@rhul.ac.uk
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Abstract

A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later, in 1921, he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this paper we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.

MSC classification

Primary: 20B05: General theory for finite groups
Secondary: 20B15: Primitive groups 20C15: Ordinary representations and characters
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 613 - 633
Copyright
Copyright © Cambridge Philosophical Society 2019

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References

REFERENCES

Aschbacher, M.. On the maximal subgroups of the finite classical groups. Invent. Math. 76 (1984), 469–514.CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playoust, C.. The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. Computational Algebra and Number Theory (London, 1993).CrossRefGoogle Scholar
Burnside, W.. On some properties of groups of odd order. Proc. London Math. Soc. 33 (1901), 162–185.Google Scholar
Burnside, W.. On simply transitive groups of prime degree. Q. J. Math. 37 (1906), 215–221.Google Scholar
Burnside, W.. On certain simply-transitive permutation-groups. Proc. Camb. Phil. Soc. 20 (1921), 482–484.Google Scholar
Burnside, W.. Theory of Groups of Finite Order (Dover Publications Inc., New York, 1955) (reprint of 2nd edition, Cambridge University Press, 1911).Google Scholar
Burnside, W.. The collected papers of William Burnside. Vol. I (Oxford University Press, Oxford, 2004). Commentary on Burnside’s life and work; papers 1883–1899, edited by Peter, M. Neumann, Mann, A. J. S. and Julia, C. Tompson, with a preface by Neumann and Mann.Google Scholar
Burnside, W.. The collected papers of William Burnside. Vol. II(Oxford University Press, Oxford, 2004). papers 1900–1926, edited by Peter, M. Neumann, Mann, A. J. S. and Julia, C. Tompson, with a preface by Neumann and Mann.Google Scholar
Cameron, P. J. and Kantor, W. M.. 2-transitive and antiflag transitive collineation groups of finite projective spaces, J. Algebra 60 (1979), 384–422.CrossRefGoogle Scholar
Caranti, A., Volta, F. D. and Sala, M.. Abelian regular subgroups of the affine group and radical rings, Publ. Math. Debrecen 69 (2006), 297–308.Google Scholar
Clifford, A. H.. Representations induced in an invariant subgroup, Ann. of Math. (2) 38 (1937), 533–550.CrossRefGoogle Scholar
Curtis, C. W., Kantor, W.M. and Seitz, G. M.. The 2-transitive permutation representations of the finite Chevalley groups. Trans. Amer. Math. Soc. 218 (1976), 1–59.Google Scholar
Dixon, J. D. and Mortimer, B.. Permutation groups. Graduate Texts in Math. vol. 163 (Springer-Verlag, New York, 1996).CrossRefGoogle Scholar
Feit, W.. Some consequences of the classification of finite simple groups. The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979). Proc. Sympos. Pure Math., vol. 37 (Amer. Math. Soc., Providence, R.I., 1980), pp. 175–181.CrossRefGoogle Scholar
The GAP Group. GAP – Groups, Algorithms and Programming, Version 4·7·4 (2014).Google Scholar
Hering, C.. Transitive linear groups and linear groups which contain irreducible subgroups of prime order. Geom. Dedicata 2 (1974), 425–460.CrossRefGoogle Scholar
Hering, C.. Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II J. Algebra 93 (1985), 151–164.CrossRefGoogle Scholar
Hiss, G. and Malle, G.. Low-dimensional representations of quasi-simple groups, LMS J. Comput. Math. 4 (2001), 22–63.CrossRefGoogle Scholar
Hiss, G. and Malle, G.. Corrigenda: Low-dimensional representations of quasi-simple groups, LMS J. Comput. Math. 5 (2002), 95–126.CrossRefGoogle Scholar
James, G. and Kerber, A.. The representation theory of the symmetric group. Encyclopedia Math. Appli. vol. 16 (Addison-Wesley Publishing Co., Reading, Mass., 1981).Google Scholar
James., G. D.The representation theory of the symmetric groups Lecture Notes in Math. vol. 682 (Springer, Berlin, 1978).CrossRefGoogle Scholar
Jansen, C., Lux, K., Parker, R. and Wilson, R.. An atlas of Brauer characters. London Math. Soc. Monogr. NewSeries, vol. 11 (The Clarendon Press, Oxford University Press, New York, 1995). Appendix 2 by T. Breuer and S. Norton (Oxford Science Publications).Google Scholar
Jones, G. A.. Cyclic regular subgroups of primitive permutation groups. J. Group Theory 5 (2002), 403–407.CrossRefGoogle Scholar
Kantor, W. M.. Some consequences of the classification of finite simple groups. Finite groups—coming of age (Montreal, Que., 1982) Contemp. Math., vol. 45 (Amer. Math. Soc., Providence, RI, 1985), pp. 159–173.Google Scholar
Knapp, W.. On Burnside’s method. J. Algebra 175 (1995), 644–660.CrossRefGoogle Scholar
KochendÖRFFER, R.. Untersuchungen über eine Vermutung von W. Burnside. Schriften des mathematischen Seminars und des Instituts für angewandtemathematik der Universität Berlin. 3 (1937), 155–180.Google Scholar
Li., C. H.The finite primitive permutation groups containing an abelian regular subgroup. Proc. London Math. Soc. (3) 87 (2003), 725–747.CrossRefGoogle Scholar
Li, C. H.. Permutation groups with a cyclic regular subgroup and arc transitive circulants. J. Algebraic Combin 21 (2005), 131–136.CrossRefGoogle Scholar
Liebeck, M. W., Praeger, C. E. and Saxl, J.. Transitive subgroups of primitive permutation groups. J. Algebra 234 (2000), 291–361.CrossRefGoogle Scholar
Lübeck, F.. Small degree representations of finite Chevalley groups in defining characteristic. LMS J. Comput. Math. 4 (2001), 135–169.Google Scholar
Manning, D.. On simply transitive groups with transitive abelian subgroups of the same degree. Trans. Amer. Math. Soc. 40 (1936), 324–342.CrossRefGoogle Scholar
Jones, S. P.et al. The Haskell 98 language and libraries: The revised report J. Functional Programming 13 (2003), 0–255, http://www.haskell.org/definition/.Google Scholar
Ramanujan, S., On certain trigonometrical sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259–276.Google Scholar
Schur, I.. Neuer Beweis eines Satzes von W. Burnside. Jahresbericht der Deutschen Mathematik-Vereinigung 17 (1908).Google Scholar
Tamaschke., O.Zur Theorie der Permutationsgruppen mit regulärer Untergruppe. I. Math. Zeit. 80 (1963) 328–354.CrossRefGoogle Scholar
Washington, L. C.. Introduction to cyclotomic fields, second ed. Graduate Texts in Math, vol. 83 (Springer-Verlag, New York, 1997).Google Scholar
Wielandt, H.. Zur Theorie der einfach transitiven Permutationsgruppen. Math. Z. 40 (1936), 582–587.CrossRefGoogle Scholar
Wielandt, H.. Finite Permutation Groups. Translated from the German by R. Bercov (Academic Press, New York-London, 1964).Google Scholar
Wielandt, H.. Mathematische Werke/Mathematical works. Vol. 1 (Walter de Gruyter & Co., Berlin, 1994). Group theory with essays on some of Wielandt’s works by G. Betsch, B. Hartley, I. M. Isaacs, O. H. Kegel and P. M. Neumann Edited with a preface by Bertram Huppert and Hans Schneider.Google Scholar
Zsigmondy, K.. Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3 (1892), 265–284.CrossRefGoogle Scholar