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On a question of B. H. Neumann


Authors: Robert Guralnick and Igor Pak
Journal: Proc. Amer. Math. Soc. 131 (2003), 2021-2025
MSC (2000): Primary 20D60
DOI: https://doi.org/10.1090/S0002-9939-02-06752-7
Published electronically: December 30, 2002
MathSciNet review: 1963745
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Abstract: The automorphism group of a free group $\mathrm{Aut}(F_k)$ acts on the set of generating $k$-tuples $(g_1,\dots,g_k)$ of a group $G$. Higman showed that when $k=2$, the union of conjugacy classes of the commutators $[g_1,g_2]$ and $[g_2,g_1]$ is an orbit invariant. We give a negative answer to a question of B.H. Neumann, as to whether there is a generalization of Higman's result for $k \ge 3$.


References [Enhancements On Off] (What's this?)

  • [C] R. Carter, Simple groups of Lie type, Reprint of the 1972 original. Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. MR 90g:20001
  • [CP] G. Cooperman, I. Pak, The product replacement graph on generating triples of permutations, preprint, 2000.
  • [D1] M. J. Dunwoody, On $T$-systems of groups, J. Austral. Math. Soc. 3 (1963), 172-179. MR 27:3706
  • [D2] M. J. Dunwoody, Nielsen Transformations, in: Computational Problems in Abstract Algebra, 45-46, Pergamon, Oxford, 1970. MR 41:5472
  • [E1] M. Evans, Ph. D. Thesis, University of Wales, 1985.
  • [E2] M. Evans, $T$-systems of certain finite simple groups, Math. Proc. Cambridge Philos. Soc. 113 (1993), 9-22. MR 93m:20022
  • [G] R. Gilman, Finite quotients of the automorphism group of a free group, Canad. J. Math. 29 (1977), 541-551. MR 55:8186
  • [H] F. Hausdorff, Grundzüge der Mengenlehre, Leipzig, 1914.
  • [LS] M. W. Liebeck, A. Shalev, Diameters of finite simple groups: sharp bounds and applications, Ann. of Math. (2) 154 (2001), 383-406. MR 2002m:20029
  • [N] B. H. Neumann, On a question of Gaschütz, Archiv der Math. 7 (1956), 87-90. MR 18:11e
  • [NN] B. H. Neumann, H. Neumann, Zwei Klassen charakteristischer Untergruppen und ihre Faktorgruppen, Math. Nachr. 4 (1951), 106-125. MR 12:671d
  • [P] I. Pak, What do we know about the product replacement algorithm? in: Groups and Computation III (W. Kantor, A. Seress, eds.), 301-347, deGruyter, Berlin, 2000. MR 2002d:20107
  • [PR] V. Platonov, A. Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Boston, MA, 1994. MR 95b:11039

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Additional Information

Robert Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113
Email: guralnic@math.usc.edu

Igor Pak
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307

DOI: https://doi.org/10.1090/S0002-9939-02-06752-7
Received by editor(s): June 1, 2001
Received by editor(s) in revised form: February 20, 2002
Published electronically: December 30, 2002
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2002 American Mathematical Society

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