Finite simple groups containing a self-centralizing element of order

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* .

**[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)**

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