Skip to main content
SpringerLink
Log in

How Abelian is a Finite Group?

  • Chapter
  • First Online:
The Mathematics of Paul Erdős I

  • 2236 Accesses

Summary.

The first paper with the above title was written by Erdős and Straus. Here we solve one of the problems considered there by proving that every group of order n contains an abelian subgroup of order at least \({2}^{\varepsilon \sqrt{\log n}}\) for some \(\varepsilon > 0\). This result is essentially best possible.We also give a quick survey of recent developments in related areas of group theory which were greatly stimulated by questions of Erdős.

Research partially supported by the Hungarian National Science Foundation Grants No. 4267 and No. 1909.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. S. I. Adian, The Burnside Problem and Identities in Groups, Ergebnisse der Math. vol. 95 Springer, Berlin (1979).

    Book  Google Scholar 

  2. M. Aschbacher, Finite Group Theory, Univ. Press, Cambridge (1986).

    MATH  Google Scholar 

  3. L. Babai, P. J. Cameron and P. P. Pálfy, On the orders of primitive groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161–168.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Babai, A. J. Goodman and L. Pyber, On faithful permutation representations of small degree, Comm. in Algebra 21 (1993), 1587–1602.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Bercov, On groups without abelian composition factors, J. Algebra 5 (1967), 106–109.

    Article  MathSciNet  MATH  Google Scholar 

  6. E. A. Bertram, Some applications of graph theory to finite groups, Discrete Math. 44 (1983), 31–43.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. A. Bertram, Large centralizers in finite solvable groups, Israel J. Math. 47 (1984), 335–344.

    Article  MathSciNet  MATH  Google Scholar 

  8. E. A. Bertram, Lower bounds for the number of conjugacy classes in finite solvable groups, Isr. J. Math. 75 (1991), 243–255.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. D. Blyth and D. J. S. Robinson, Recent progress on rewritability in groups, in Group Theory, Proc. Singapore Group Theory Conference 1987 (eds. K. N. Cheng and Y. K. Leong) Walter de Gruyter Berlin, New York (1988), 77–85.

    Google Scholar 

  10. R. D. Blyth and D. J. S. Robinson, Insoluble groups with P 8, J. Pure Appl. Algebra 72 (1991), 251–263.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Brandl, General bounds for permutability in finite groups, Arch. Math. 53 (1989), 245–249.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Brauer, Representations of finite groups, in Lectures in modern mathematics, Vol 1. (ed. T. L. Saaty) John Wiley and Sons, New York (1963).

    Google Scholar 

  13. R Brauer and K. A. Fowler, On groups of even order, Ann of Math. (2) 62 (1955), 565–583.

    Google Scholar 

  14. J. Buhler, R. Gupta and J. Harris, Isotropic subspaces for skewforms and maximal abelian subgroups of p-groups, J. Algebra 108 (1987), 269–279.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. A. Brodie and L. C. Kappe, Finite coverings by subgroups with a given property, Glasgow Math. J. 35 (1993), 179–188.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Carter and P. Fong, The Sylow 2-subgroups of the finite classical groups, J. Algebra 1 (1964), 139–151.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Cartwright, The order of the derived group of a BFC-group: J. London Math. Soc. (2) 30 (1984), 227–243.

    Google Scholar 

  18. A. Chermak and A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989), 907–914.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Cossey, Finite soluble groups have large centralisers, Bull. Aust. Math. Soc. 35 (1987), 291–298.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. D. Dixon, The Fitting subgroup of a linear solvable group, J. Austral. Math. Soc. 7 (1967), 417–424.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. D. Dixon, Maximal abelian subgroups of the symmetric groups, Can. J. Math. XXIII (1971),426–438.

    Google Scholar 

  22. L. Dornhoff, Group representation theory, Part A, Dekker, New York (1972).

    MATH  Google Scholar 

  23. P. Erdős, On some problems in graph theory, combinatorial analysis and combinatorial number theory, in Graph theory and Combinatorics, Acad. Press, London (1984), 1–17.

    Google Scholar 

  24. P. Erdős, Some of my favourite unsolved problems, in A Tribute to Paul Erdős (eds. A. Baker, B. Bollobás and A. Hajnal), Cambridge Univ, Press (1990), 467–478.

    Google Scholar 

  25. P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196.

    Article  MathSciNet  Google Scholar 

  26. P. Erdős and E. G. Straus, How abelian is a finite group?, Linear and Multilinear Algebra 3, Gordon and Breach (1976),307–312.

    Google Scholar 

  27. P. Erdős and P. Turán, On some problems of statistical group-theory, IV, Acta Math. Hungar. 19 (1968), 413–435.

    Article  Google Scholar 

  28. V. Faber, R. Laver and R. McKenzie, Coverings of groups by abelian subgroups, Canad. J. Math. 30 (1978), 933–945.

    Article  MathSciNet  Google Scholar 

  29. A. J. Goodman, The edge-orbit conjecture of Babai, JCT (B) 57 (1993), 26–35.

    Google Scholar 

  30. J. R. J. Groves, A conjecture of Lennox and Wiegold concerning supersoluble groups, J. Austral. Math. Soc. (A) 35 (1983), 218–220.

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc. (2) 36 (1933), 29–95.

    Google Scholar 

  32. P. Hall and C. R. Kulatilaka, A property of locally finite groups, Proc. London Math. Soc. (3) 16 (1966), 1–39.

    Google Scholar 

  33. H. Heineken, Nilpotent subgroups of finite soluble groups, Arch. Math. 56 (1991), 417–423.

    Article  MathSciNet  MATH  Google Scholar 

  34. G. Higman, B. H. Neumann and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254.

    Google Scholar 

  35. B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967.

    Book  MATH  Google Scholar 

  36. B. Huppert and N. Blackburn, Finite Groups II, Springer, Berlin, Heidelberg, New York (1981).

    Google Scholar 

  37. I. M. Isaacs, Character theory of finite groups, Acad. Press, New York (1976).

    MATH  Google Scholar 

  38. I. M. Isaacs, Solvable groups contain large centralizers, Israel J. Math. 55 (1986), 58–64.

    Article  MathSciNet  MATH  Google Scholar 

  39. L-C. Kappe, Finite coverings by 2-Engel groups, Bull. Austral. Math. Soc. 38 (1988), 141–150.

    Google Scholar 

  40. M. I. Kargapolov, On a problem of O. J. Schmidt, Sibirsk. Math, Z. 4 (1963), 232–235.

    Google Scholar 

  41. T. Kepka and M. Niemenmaa, On conjugacy classes in finite loops, Bull. Austral. Math. Soc. 38 (1988), 171–176.

    Article  MathSciNet  MATH  Google Scholar 

  42. L. G. Kovács, unpublished.

    Google Scholar 

  43. L. G. Kovács and C. R. Leedham-Green, Some normally monomial p-groups of maximal class and large derived length, Quart. J. Math. Oxford (2) 37 (1986), 49–54.

    Google Scholar 

  44. E. Landau, Über die Klassenzahl der binären quadratischen Formen von negativer Diskriminant, Math. Ann. 56 (1903), 260–270.

    Google Scholar 

  45. J. C. Lennox and J. Wiegold, Extension of a problem of Paul Erdős on groups, J. Austral. Math. Soc. (A) 31 (1981), 459–463.

    Article  MathSciNet  MATH  Google Scholar 

  46. G. A. Miller, On an important theorem with respect to the operation groups of order p α, p being any prime number, Messenger of Math. 27 (1898), 119–121.

    Google Scholar 

  47. I. D. Macdonald, Some explicit bounds in groups, Proc. London Math. Soc. (3) 11 (1969), 23–56.

    Google Scholar 

  48. A. Mann, Some applications of powerful p-groups, Proc. Groups St. Andrews 1989, Cambridge (1991), 370–385.

    Google Scholar 

  49. G. Zh. Mantashyan, The number of generators of finite p-groups and nilpotent groups without torsion and dimensions of associative rings and Lie algebras (in Russian) Matematika 6 (1988), 178–186, Zbl. Math. 744.20034.

    Google Scholar 

  50. U. Martin, Almost all p-groups have automorphism group, a p-group, Bull. Amer. Math. Soc. 15 (1986), 78–82.

    Article  MathSciNet  MATH  Google Scholar 

  51. D. R. Mason, On coverings of groups by abelian subgroups, Math. Proc. Cambridge Phil. Soc. 83 (1978), 205–209.

    Article  MATH  Google Scholar 

  52. B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236–248.

    Article  MathSciNet  MATH  Google Scholar 

  53. B. H. Neumann, Groups covered by finitely many cosets, Publ. Math. Debrecen 3 (1954), 227–242.

    MathSciNet  MATH  Google Scholar 

  54. B. H. Neumann, A problem of Paul Erdős on groups, J. Austral. Math. Soc. 21 (1976), 467–472.

    Article  MATH  Google Scholar 

  55. P. M. Neumann, Two combinatorial problems in group theory, Bull. London Math. Soc. 21 (1989), 456–458.

    Article  MathSciNet  MATH  Google Scholar 

  56. P. M. Neumann and M. R. Vaughan-Lee, An essay on BFC groups, Proc. London Math. Soc. (3) 35 (1977), 213–237.

    Google Scholar 

  57. A. Yu. Ol’shanskii, The number of generators and orders of abelian subgroups of finite p-groups, Math. Notes 23 (1978), 183–185.

    Google Scholar 

  58. A. Yu. Ol’shanskii, Geometry of defining relations in groups, Kluwer, Dordrecht (1991).

    Google Scholar 

  59. P. P. Pálfy, A polynomial bound for the orders of primitive solvable groups, J. Algebra 77 (1982), 127–137.

    Article  MathSciNet  MATH  Google Scholar 

  60. L. Pyber, The number of pairwise non-commuting elements and the index of the center in a finite group, J. London Math. Soc. (2) 35 (1987), 287–295.

    Google Scholar 

  61. L. Pyber, Finite groups have many conjugacy classes, J. London Math. Soc. (2) 46 (1992), 239–249.

    Google Scholar 

  62. E. Rips, Generalized small cancellation theory and applications II (unpublished).

    Google Scholar 

  63. D. J. S. Robinson, Finiteness, Solubility and Nilpotence, in Group Theory essays for Philip Hall (eds. K. W. Gruenberg and J. E. Roseblade) Acad. Press, London (1984), 159–206.

    Google Scholar 

  64. G. R. Robinson, On linear groups, J. Algebra 131 (1990), 527–534.

    Article  MathSciNet  MATH  Google Scholar 

  65. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.

    MathSciNet  MATH  Google Scholar 

  66. S. Shelah, On the number of non-conjugate subgroups, Algebra Universalis 16 (1983), 131–146.

    Article  MathSciNet  MATH  Google Scholar 

  67. M. Suzuki, Group Theory I, II, Springer, New York, 1986.

    Book  Google Scholar 

  68. M. J. Tomkinson, FC-groups, Research notes in mathematics 96, Pitman, London (1984).

    Google Scholar 

  69. M. J. Tomkinson, Groups covered by abelian subgroups, Proc. Groups St. Andrews 1985, Cambridge (1986), 332–334.

    Google Scholar 

  70. M. J. Tomkinson, Groups covered by finitely many cosets or subgroups, Comm. in Algebra 15 (1987), 845–859.

    MathSciNet  MATH  Google Scholar 

  71. M. J. Tomkinson, Hypercentre-by-finite groups, Publ. Math. Debrecen 40 (1992), 313–321.

    MathSciNet  MATH  Google Scholar 

  72. M. R. Vaughan-Lee, Breadth and commutator subgroups of p-groups, J. Algebra 32 (1974), 278–285.

    Article  MathSciNet  MATH  Google Scholar 

  73. M. R. Vaughan-Lee and J. Wiegold, Countable locally nilpotent groups of finite exponent with no maximal subgroups, Bull. London Math. Soc. 13 (1981), 45–46.

    Article  MathSciNet  MATH  Google Scholar 

  74. E. I. Zelmanov, On the Restricted Burnside Problem, in Proc. Int. Congress of Math. Kyoto, Japan 1990, Springer, Tokyo (1991), 1479–1489.

    Google Scholar 

  75. J. Wiegold, Groups with boundedly finite classes of conjugate elements, Proc. Roy. Soc. London (A) 238 (1956), 389–401.

    Article  MathSciNet  Google Scholar 

  76. J. S. Wilson, Two-generator conditions for residually finite groups, Bull. London Math. Soc. 23 (1991), 239–248.

    Article  MathSciNet  MATH  Google Scholar 

  77. T. R. Wolf, Solvable and nilpotent subgroups of GL(n, q m), Canad. J. Math. 34 (1982), 1097–1111.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, 127, Budapest, H-1364, Hungary

    Lásló Pyber

Authors
  1. Lásló Pyber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lásló Pyber .

Editor information

Editors and Affiliations

  1. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

  2. Department of Applied Mathematics, Charles University Department of Applied Mathematics, Prague, Czech Republic

    Jaroslav Nešetřil

  3. Department of Mathematics, Iowa State University, Ames, Iowa, USA

    Steve Butler

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Pyber, L. (2013). How Abelian is a Finite Group?. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_25

Download citation

Publish with us

Policies and ethics

Search

Navigation