How Abelian is a Finite Group?

  • Lásló Pyber


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.


  1. 1.
    S. I. Adian, The Burnside Problem and Identities in Groups, Ergebnisse der Math. vol. 95 Springer, Berlin (1979).CrossRefGoogle Scholar
  2. 2.
    M. Aschbacher, Finite Group Theory, Univ. Press, Cambridge (1986).MATHGoogle Scholar
  3. 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.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    L. Babai, A. J. Goodman and L. Pyber, On faithful permutation representations of small degree, Comm. in Algebra 21 (1993), 1587–1602.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    R. Bercov, On groups without abelian composition factors, J. Algebra 5 (1967), 106–109.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    E. A. Bertram, Some applications of graph theory to finite groups, Discrete Math. 44 (1983), 31–43.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    E. A. Bertram, Large centralizers in finite solvable groups, Israel J. Math. 47 (1984), 335–344.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    E. A. Bertram, Lower bounds for the number of conjugacy classes in finite solvable groups, Isr. J. Math. 75 (1991), 243–255.MathSciNetMATHCrossRefGoogle Scholar
  9. 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. 10.
    R. D. Blyth and D. J. S. Robinson, Insoluble groups with P 8, J. Pure Appl. Algebra 72 (1991), 251–263.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    R. Brandl, General bounds for permutability in finite groups, Arch. Math. 53 (1989), 245–249.MathSciNetMATHCrossRefGoogle Scholar
  12. 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. 13.
    R Brauer and K. A. Fowler, On groups of even order, Ann of Math. (2) 62 (1955), 565–583.Google Scholar
  14. 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.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    M. A. Brodie and L. C. Kappe, Finite coverings by subgroups with a given property, Glasgow Math. J. 35 (1993), 179–188.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    R. Carter and P. Fong, The Sylow 2-subgroups of the finite classical groups, J. Algebra 1 (1964), 139–151.MathSciNetMATHCrossRefGoogle Scholar
  17. 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. 18.
    A. Chermak and A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989), 907–914.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    J. Cossey, Finite soluble groups have large centralisers, Bull. Aust. Math. Soc. 35 (1987), 291–298.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    J. D. Dixon, The Fitting subgroup of a linear solvable group, J. Austral. Math. Soc. 7 (1967), 417–424.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    J. D. Dixon, Maximal abelian subgroups of the symmetric groups, Can. J. Math. XXIII (1971),426–438.Google Scholar
  22. 22.
    L. Dornhoff, Group representation theory, Part A, Dekker, New York (1972).MATHGoogle Scholar
  23. 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. 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. 25.
    P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196.MathSciNetCrossRefGoogle Scholar
  26. 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. 27.
    P. Erdős and P. Turán, On some problems of statistical group-theory, IV, Acta Math. Hungar. 19 (1968), 413–435.CrossRefGoogle Scholar
  28. 28.
    V. Faber, R. Laver and R. McKenzie, Coverings of groups by abelian subgroups, Canad. J. Math. 30 (1978), 933–945.MathSciNetCrossRefGoogle Scholar
  29. 29.
    A. J. Goodman, The edge-orbit conjecture of Babai, JCT (B) 57 (1993), 26–35.Google Scholar
  30. 30.
    J. R. J. Groves, A conjecture of Lennox and Wiegold concerning supersoluble groups, J. Austral. Math. Soc. (A) 35 (1983), 218–220.MathSciNetMATHCrossRefGoogle Scholar
  31. 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. 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. 33.
    H. Heineken, Nilpotent subgroups of finite soluble groups, Arch. Math. 56 (1991), 417–423.MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    G. Higman, B. H. Neumann and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254.Google Scholar
  35. 35.
    B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967.MATHCrossRefGoogle Scholar
  36. 36.
    B. Huppert and N. Blackburn, Finite Groups II, Springer, Berlin, Heidelberg, New York (1981).Google Scholar
  37. 37.
    I. M. Isaacs, Character theory of finite groups, Acad. Press, New York (1976).MATHGoogle Scholar
  38. 38.
    I. M. Isaacs, Solvable groups contain large centralizers, Israel J. Math. 55 (1986), 58–64.MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    L-C. Kappe, Finite coverings by 2-Engel groups, Bull. Austral. Math. Soc. 38 (1988), 141–150.Google Scholar
  40. 40.
    M. I. Kargapolov, On a problem of O. J. Schmidt, Sibirsk. Math, Z. 4 (1963), 232–235.Google Scholar
  41. 41.
    T. Kepka and M. Niemenmaa, On conjugacy classes in finite loops, Bull. Austral. Math. Soc. 38 (1988), 171–176.MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    L. G. Kovács, unpublished.Google Scholar
  43. 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. 44.
    E. Landau, Über die Klassenzahl der binären quadratischen Formen von negativer Diskriminant, Math. Ann. 56 (1903), 260–270.Google Scholar
  45. 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.MathSciNetMATHCrossRefGoogle Scholar
  46. 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. 47.
    I. D. Macdonald, Some explicit bounds in groups, Proc. London Math. Soc. (3) 11 (1969), 23–56.Google Scholar
  48. 48.
    A. Mann, Some applications of powerful p-groups, Proc. Groups St. Andrews 1989, Cambridge (1991), 370–385.Google Scholar
  49. 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. 50.
    U. Martin, Almost all p-groups have automorphism group, a p-group, Bull. Amer. Math. Soc. 15 (1986), 78–82.MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    D. R. Mason, On coverings of groups by abelian subgroups, Math. Proc. Cambridge Phil. Soc. 83 (1978), 205–209.MATHCrossRefGoogle Scholar
  52. 52.
    B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236–248.MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    B. H. Neumann, Groups covered by finitely many cosets, Publ. Math. Debrecen 3 (1954), 227–242.MathSciNetMATHGoogle Scholar
  54. 54.
    B. H. Neumann, A problem of Paul Erdős on groups, J. Austral. Math. Soc. 21 (1976), 467–472.MATHCrossRefGoogle Scholar
  55. 55.
    P. M. Neumann, Two combinatorial problems in group theory, Bull. London Math. Soc. 21 (1989), 456–458.MathSciNetMATHCrossRefGoogle Scholar
  56. 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. 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. 58.
    A. Yu. Ol’shanskii, Geometry of defining relations in groups, Kluwer, Dordrecht (1991).Google Scholar
  59. 59.
    P. P. Pálfy, A polynomial bound for the orders of primitive solvable groups, J. Algebra 77 (1982), 127–137.MathSciNetMATHCrossRefGoogle Scholar
  60. 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. 61.
    L. Pyber, Finite groups have many conjugacy classes, J. London Math. Soc. (2) 46 (1992), 239–249.Google Scholar
  62. 62.
    E. Rips, Generalized small cancellation theory and applications II (unpublished).Google Scholar
  63. 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. 64.
    G. R. Robinson, On linear groups, J. Algebra 131 (1990), 527–534.MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.MathSciNetMATHGoogle Scholar
  66. 66.
    S. Shelah, On the number of non-conjugate subgroups, Algebra Universalis 16 (1983), 131–146.MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    M. Suzuki, Group Theory I, II, Springer, New York, 1986.CrossRefGoogle Scholar
  68. 68.
    M. J. Tomkinson, FC-groups, Research notes in mathematics 96, Pitman, London (1984).Google Scholar
  69. 69.
    M. J. Tomkinson, Groups covered by abelian subgroups, Proc. Groups St. Andrews 1985, Cambridge (1986), 332–334.Google Scholar
  70. 70.
    M. J. Tomkinson, Groups covered by finitely many cosets or subgroups, Comm. in Algebra 15 (1987), 845–859.MathSciNetMATHGoogle Scholar
  71. 71.
    M. J. Tomkinson, Hypercentre-by-finite groups, Publ. Math. Debrecen 40 (1992), 313–321.MathSciNetMATHGoogle Scholar
  72. 72.
    M. R. Vaughan-Lee, Breadth and commutator subgroups of p-groups, J. Algebra 32 (1974), 278–285.MathSciNetMATHCrossRefGoogle Scholar
  73. 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.MathSciNetMATHCrossRefGoogle Scholar
  74. 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. 75.
    J. Wiegold, Groups with boundedly finite classes of conjugate elements, Proc. Roy. Soc. London (A) 238 (1956), 389–401.MathSciNetCrossRefGoogle Scholar
  76. 76.
    J. S. Wilson, Two-generator conditions for residually finite groups, Bull. London Math. Soc. 23 (1991), 239–248.MathSciNetMATHCrossRefGoogle Scholar
  77. 77.
    T. R. Wolf, Solvable and nilpotent subgroups of GL(n, q m), Canad. J. Math. 34 (1982), 1097–1111.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

Personalised recommendations