Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-17T07:38:28.411Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-transitive actions on conjugacy classes

Published online by Cambridge University Press:  17 April 2009

Michael J.J. Barry  and
Michael B. Ward
Michael J.J. Barry
Affiliation:
Department of MathematicsAllegheny CollegeMeadville PA 16335, United States of America e-mail: mbarry@alleg.edu
Michael B. Ward
Affiliation:
Department of MathematicsWestern Oregon UniversityMonmouth OR 97361, United States of America e-mail: wardm@wou.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Every group acts transitively by conjugation on each of its conjugacy classes of elements. It is natural to wonder when this action becomes multiply transitive. In this paper, we determine all finite groups which act faithfully and 2-transitively on a conjugacy class of elements. We also give some consequences including a solvability criterion based on what fraction of elements belong to conjugacy classes upon which the group acts faithfully and 2–transitively.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 3 , June 1999 , pp. 409 - 420
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Brewster, B. and Ward, M.B., ‘Groups 2–transitive on a set of their Sylow subgroups’, Bull. Austral. Math. Soc. 52 (1995), 117–136.CrossRefGoogle Scholar
[2]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.CrossRefGoogle Scholar
[3]Dieudonné, J.A., Le géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete 5 (Springer-Verlag, Berlin, Heidelberg, New York, 1955).Google Scholar
[4]Dixon, J.D. and Mortimer, B., Permutation groups, Graduate Texts in Mathematics 163 (Springer-Verlag, Berlin, Heidelberg, New York, 1996).CrossRefGoogle Scholar
[5]Hering, C., ‘Transitive linear groups and linear groups which contain irreducible subgroups of prime order’, Geom. Dedicata 2 (1974), 425–460.Google Scholar
[6]Hering, C., ‘Transitive linear groups and linear groups which contain irreducible subgroups of prime order, II’, J. Algebra 93 (1985), 151–164.CrossRefGoogle Scholar
[7]Huppert, B., ‘Zweifach transitive, auflösbare Permutationgruppen’, Math. Z. 68 (1957), 126–150.CrossRefGoogle Scholar
[8]Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, Heidleberg, New York, 1967).CrossRefGoogle Scholar
[9]Huppert, B. and Blackburn, N., Finite groups II (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[10]Kleidman, P. and Liebeck, M., The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series 129 (Cambridge University Press, Cambridge, New York, 1990).Google Scholar
[11]Maillet, E., ‘Sur les isomorphes holoédriques et transitifs des groupes symétrique ou alternés’, J. Math. Pures Appl. 1 (1895), 5–34.Google Scholar
[12]O'Meara, O.T., Lectures on linear groups, Regional Conference Series in Mathematics 22 (American Mathematical Society, Providence, RI, 1974).Google Scholar
[13]O'Meara, O.T., Symplectic groups, Mathematical Surveys 16 (American Mathematical Society, Providence, RI, 1978).CrossRefGoogle Scholar
[14]Playtis, A., Sehgal, S. and Zassenhaus, H., ‘Equidistributed permutation groups’, Comm. Algebra 6 (1978), 35–57.CrossRefGoogle Scholar
[15]Steinberg, R., Lectures on Chevalley Groups (Department of Mathematics, Yale University, 1967).Google Scholar