Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Finite simple groups containing a self-centralizing element of order $ 6$


Authors: John L. Hayden and David L. Winter
Journal: Proc. Amer. Math. Soc. 66 (1977), 202-204
MSC: Primary 20D05
DOI: https://doi.org/10.1090/S0002-9939-1977-0450393-1
MathSciNet review: 0450393
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By a self-centralizing element of a group we mean an element which commutes only with its powers. In this paper we establish the following result:

Theorem. Let G be a finite simple group which has a self-centralizing element of order 6. Assume that G has only one class of involutions. Then G is isomorphic to one of the groups $ {M_{11}},{J_1},{L_3}(3),{L_2}(11),{L_2}(13)$.


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

  • [1] M. Aschbacher, Thin finite simple groups, Bull. Amer. Math. Soc. 82 (1976), 484. MR 0396735 (53:596)
  • [2] W. Feit and J. G. Thompson, Finite groups which contain a self-centralizing subgroup of order 3, Nagoya Math. J. 21 (1962), 185-197. MR 0142623 (26:192)
  • [3] D. Gorenstein, Centralizers of involution of finite simple groups, Finite Simple Groups, Academic Press, London, 1971. MR 0335622 (49:402)
  • [4] -, Finite groups, Harper and Row, New York, 1968. MR 0231903 (38:229)
  • [5] D. Gorenstein and K. Harada, Finite groups whose 2-subgroups are generated by at most 4 elements, Mem. Amer. Math. Soc. No. 147, 1974. MR 0367048 (51:3290)
  • [6] G. Higman, Odd characterizations of finite simple groups, (Lecture notes, University of Michigan, 1968).
  • [7] Z. Janko, A new finite simple group with Abelian Sylow 2-subgroups and its characterization, J. Algebra 3 (1966), 147-186. MR 0193138 (33:1359)
  • [8] -, A class of simple groups of characteristic 2, Finite Groups' 72, North-Holland, New York, 1973. MR 0360800 (50:13247)
  • [9] Z. Janko and J. G. Thompson, On a class of finite simple groups of Ree, J. Algebra 4 (1966), 274-292. MR 0201504 (34:1386)
  • [10] L. Schiefelbusch, Sylow 2-subgroups of simple groups, J. Algebra 31 (1974), 131-153. MR 0357599 (50:10067)
  • [11] I. Schur, Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. für die r. und a Math. 132 (1907), 85-137.
  • [12] M. Suzuki, On finite groups containing an element of order four which commutes only with its powers, Illinois J. Math. 3 (1959), 255-271. MR 0104733 (21:3486)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20D05

Retrieve articles in all journals with MSC: 20D05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0450393-1
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society