The Ore conjecture

The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Despite considerable effort, it remains open for various infinite families of simple groups. In this paper we develop new strategies, combining character theoretic methods with other ingredients, and use them to establish the conjecture. Liebeck acknowledges the support of a Maclaurin Fellowship from the New Zealand Institute of Mathematics and its Applications. O’Brien acknowledges the support of an LMS Visitor Grant, the Marsden Fund of New Zealand (grant UOA 0721), and the Mathematical Sciences Research Institute (Berkeley). Shalev acknowledges the support of an EPSRC Visiting Fellowship, an Israel Science Foundation Grant, and a Bi-National Science Foundation grant United States-Israel 2004-052. Tiep acknowledges the support of the NSF (grant DMS-0600967), and the Mathematical Sciences Research Institute (Berkeley).

[1]  O. Ore.  SOME REMARKS ON COMMUTATORS , 1951 .

[2]  M. Marcus,et al.  On Matrix Commutators , 1960, Canadian Journal of Mathematics.

[3]  Commutators in the special and general linear groups , 1961 .

[4]  Commutators of matrices with coefficients from the field of two elements , 1962 .

[5]  G. E. Wall.  On the conjugacy classes in the unitary, symplectic and orthogonal groups , 1963, Journal of the Australian Mathematical Society.

[6]  Bomshik Chang,et al.  The conjugate classes of Chevalley groups of type (G2) , 1968 .

[7]  T. A. Springer,et al.  Seminar on Algebraic Groups and Related Finite Groups , 1970 .

[8]  Michael Aschbacher,et al.  Corrections to “Involutions in Chevalley groups over fields of even order” , 1976, Nagoya Mathematical Journal.

[9]  P. Gérardin,et al.  Weil representations associated to finite fields , 1977 .

[10]  G. Lusztig,et al.  The characters of the finite unitary groups , 1977 .

[11]  Charles W. Curtis,et al.  Representations of finite groups of Lie type , 1979 .

[12]  W. Hesselink.  Nilpotency in classical groups over a field of characteristic 2 , 1979 .

[13]  Kenzo Mizuno.  The Conjugate Classes of Unipotent Elements of the Chevalley Groups $E_7$ and $E_8$ , 1980 .

[14]  D. Deriziotis.  The Centralizers of Semisimple Elements of the Chevalley Groups $E_7$ and $E_8$ , 1983 .

[15]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[16]  Roger W. Carter,et al.  Finite groups of Lie type: Conjugacy classes and complex characters , 1985 .

[17]  Commutators in the symplectic group , 1988 .

[18]  Martin W. Liebeck,et al.  The Subgroup Structure of the Finite Classical Groups , 1990 .

[19]  Über Kommutatoren in endlichen einfachen Gruppen , 1992 .

[20]  Gary M. Seitz,et al.  SUBGROUPS OF MAXIMAL RANK IN FINITE EXCEPTIONAL GROUPS OF LIE TYPE , 1992 .

[21]  I. Janiszczak,et al.  The semisimple conjugacy classes of finite groups of lie type E6 and E7 , 1993 .

[22]  Gary M. Seitz,et al.  Subgroups generated by root elements in groups of Lie type , 1994 .

[23]  P. Fleischmann,et al.  The semisimple conjugacy classes and the generic class number of the finite simple groups of lie type E8 , 1994 .

[24]  H. I. Blau.  A fixed-point theorem for central elements in quasisimple groups , 1994 .

[25]  Scott H. Murray,et al.  Generating random elements of a finite group , 1995 .

[26]  D. Gluck.  Sharper Character Value Estimates for Groups of Lie Type , 1995 .

[27]  M. Liebeck,et al.  Reductive subgroups of exceptional algebraic groups , 1996 .

[28]  P. Tiep,et al.  Minimal characters of the finite classical groups , 1996 .

[29]  P. Tiep,et al.  Some Characterizations of the Weil Representations of the Symplectic and Unitary Groups , 1997 .

[30]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[31]  R. Gow.  Commutators in Finite Simple Groups of Lie Type , 2000 .

[32]  Frank Lübeck.  SMALLEST DEGREES OF REPRESENTATIONS OF EXCEPTIONAL GROUPS OF LIE TYPE , 2001 .

[33]  R. Guralnick,et al.  Cross characteristic representations of even characteristic symplectic groups , 2004 .

[34]  M. Liebeck,et al.  Character Degrees and Random Walks in Finite Groups of Lie Type , 2005 .

[35]  Rank 3 permutation modules of the finite classical groups , 2005 .

[36]  William R. Unger,et al.  Computing the character table of a finite group , 2006, J. Symb. Comput..

[37]  Derek F. Holt,et al.  Computing conjugacy class representatives in permutation groups , 2006 .

[38]  Meinolf Geck,et al.  Group representation theory , 2007 .

[39]  A. Shalev.  Word maps, conjugacy classes, and a noncommutative Waring-type theorem , 2009 .

[40]  Shelly Garion,et al.  Commutator maps, measure preservation, and T-systems , 2007, 0708.3664.

[41]  R. Guralnick,et al.  Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements , 2009, 0902.2238.