Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T16:05:20.624Z Has data issue: false hasContentIssue false
  • English
  • Français

The general theory of 3-transposition groups

Published online by Cambridge University Press:  24 October 2008

J. I. Hall
J. I. Hall
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

A set D of 3-transpositions in the group G is a normal set of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. The study of 3-transposition groups was instituted by Bernd Fischer [6, 7, 8] who classified all 3-transposition groups which are finite and have no non-trivial normal solvable subgroups. Recently the present author and H. Cuypers[5] extended Fischer's result to include all 3-transposition groups with trivial centre. For this classification the present paper provides the extension of Fischer's paper [8] where he gave two basic reductions, the Normal Subgroup Theorem and the Transitivity Theorem stated below. Other results of help in the classification are also presented here.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 2 , September 1993 , pp. 269 - 294
Copyright
Copyright © Cambridge Philosophical Society 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aschbacher, M.. A homomorphism theorem for finite graphs. Proc. Amer. Math. Soc. 54 (1976), 468–470.CrossRefGoogle Scholar
[2]Aschbacher, M.. Finite Group Theory. Cambridge University Press, 1986.Google Scholar
[3]Bruck, R. H.. A Survey of Binary Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete 20, Springer-Verlag, 1958.CrossRefGoogle Scholar
[4]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.. Atlas of Finite Groups. Clarendon Press, 1985.Google Scholar
[5]Cuypers, H. and Hall, J. I.. The classification of 3-transposition groups with trivial center. In: Groups, Combinatorics and Geometry, Durham 1990, eds. Liebeck, M. and Saxl, J., London Math. Soc. Lecture Notes 165 (1992), Cambridge University Press, 121–138.Google Scholar
[6]Fischer, B.. A characterization of the symmetric groups on 4 and 5 letters. J. Algebra 3 (1966), 88–98.CrossRefGoogle Scholar
[7]Fischer, B.. Finite groups generated by 3-transpositions. University of Warwick lecture notes, 1969.Google Scholar
[8]Fischer, B.. Finite groups generated by 3-transpositions I. Invent. Math. 13 (1971), 232–246.CrossRefGoogle Scholar
[9]Hall, J. I.. 3-Transposition groups with non-central normal 2-subgroups. J. Algebra 146 (1992), 49–76.CrossRefGoogle Scholar
[10]Hall, M. Jr. Automorphisms of Steiner triple systems. Proc. Symp. in Pure Math. 6 (1962), 47–66.CrossRefGoogle Scholar
[11]Hall, M. Jr. Group theory and block designs. In: Proceedings of the International Conference on the Theory of Groups at Canberra 1965, eds. Kovács, L. G. and Neumann, B. H., Gordon and Breach, 1967.Google Scholar
[12]Levi, F. W. and van der Waerden, B. L.. Über eine besondere Klasse von Gruppen. Abh. Math. Sem. Hamburg 9 (1933), 154–158.Google Scholar
[13]Weiss, R.. 3-Transpositions in infinite groups. Math. Proc. Cambridge Philos. Soc. 96 (1984), 371–377.CrossRefGoogle Scholar
[14]Zara, F.. Classification des couples fischeriens. Thése, Université de Picardie, 1984.Google Scholar
[15]Zara, F.. A first step toward the classification of Fischer groups. Geom. Dedicata 25 (1988), 503–512.CrossRefGoogle Scholar