Finite simple groups containing a selfcentralizing element of order
Authors:
John L. Hayden and David L. Winter
Journal:
Proc. Amer. Math. Soc. 66 (1977), 202204
MSC:
Primary 20D05
MathSciNet review:
0450393
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: By a selfcentralizing 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 selfcentralizing element of order 6. Assume that G has only one class of involutions. Then G is isomorphic to one of the groups .
 [1]
Michael
Aschbacher, Thin finite simple groups, Bull. Amer. Math. Soc. 82 (1976), no. 3, 484. MR 0396735
(53 #596), http://dx.doi.org/10.1090/S000299041976140633
 [2]
Walter
Feit and John
G. Thompson, Finite groups which contain a selfcentralizing
subgroup of order 3., Nagoya Math. J. 21 (1962),
185–197. MR 0142623
(26 #192)
 [3]
D.
Gorenstein, Centralizers of involutions in finite simple
groups, Finite simple groups (Proc. Instructional Conf., Oxford, 1969)
Academic Press, London, 1971, pp. 65–133. MR 0335622
(49 #402)
 [4]
Daniel
Gorenstein, Finite groups, Harper & Row, Publishers, New
YorkLondon, 1968. MR 0231903
(38 #229)
 [5]
Daniel
Gorenstein and Koichiro
Harada, Finite groups whose 2subgroups are generated by at most 4\
elements, American Mathematical Society, Providence, R.I., 1974.
Memoirs of the American Mathematical Society, No. 147. MR 0367048
(51 #3290)
 [6]
G. Higman, Odd characterizations of finite simple groups, (Lecture notes, University of Michigan, 1968).
 [7]
Zvonimir
Janko, A new finite simple group with abelian Sylow 2subgroups and
its characterization, J. Algebra 3 (1966),
147–186. MR 0193138
(33 #1359)
 [8]
Zvonimir
Janko, A class of simple groups of characteristic 2, Finite
groups ’72 (Proc. Gainesville Conf., Univ. Florida, Gainesville,
Fla., 1972) NorthHolland, Amsterdam, 1973, pp. 98–100.
NorthHolland Math. Studies, Vol. 7. MR 0360800
(50 #13247)
 [9]
Zvonimir
Janko and John
G. Thompson, On a class of finite simple groups of Ree, J.
Algebra 4 (1966), 274–292. MR 0201504
(34 #1386)
 [10]
Lary
Schiefelbusch, Sylow 2subgroups 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), 85137.
 [12]
Michio
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)
 [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 selfcentralizing subgroup of order 3, Nagoya Math. J. 21 (1962), 185197. 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 2subgroups 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 2subgroups and its characterization, J. Algebra 3 (1966), 147186. MR 0193138 (33:1359)
 [8]
 , A class of simple groups of characteristic 2, Finite Groups' 72, NorthHolland, 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), 274292. MR 0201504 (34:1386)
 [10]
 L. Schiefelbusch, Sylow 2subgroups of simple groups, J. Algebra 31 (1974), 131153. 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), 85137.
 [12]
 M. Suzuki, On finite groups containing an element of order four which commutes only with its powers, Illinois J. Math. 3 (1959), 255271. 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:
http://dx.doi.org/10.1090/S00029939197704503931
PII:
S 00029939(1977)04503931
Article copyright:
© Copyright 1977
American Mathematical Society
