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

Hayden, John L.; Winter, David L.

doi:10.1090/S0002-9939-1977-0450393-1

Finite simple groups containing a self-centralizing element of order $6$
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by John L. Hayden and David L. Winter PDF

Proc. Amer. Math. Soc. 66 (1977), 202-204 Request permission

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

Michael Aschbacher, Thin finite simple groups, Bull. Amer. Math. Soc. 82 (1976), no. 3, 484. MR 396735, DOI 10.1090/S0002-9904-1976-14063-3
Walter Feit and John G. Thompson, Finite groups which contain a self-centralizing subgroup of order 3, Nagoya Math. J. 21 (1962), 185–197. MR 142623
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
Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
Daniel Gorenstein and Koichiro Harada, Finite groups whose $2$-subgroups are generated by at most $4$ elements, Memoirs of the American Mathematical Society, No. 147, American Mathematical Society, Providence, R.I., 1974. MR 0367048

Odd characterizations of finite simple groups

Zvonimir Janko, A new finite simple group with abelian Sylow $2$-subgroups and its characterization, J. Algebra 3 (1966), 147–186. MR 193138, DOI 10.1016/0021-8693(66)90010-X
Zvonimir Janko, A class of simple groups of characteristic $2$, Finite groups ’72 (Proc. Gainesville Conf., Univ. Florida, Gainesville, Fla., 1972) North-Holland Math. Studies, Vol. 7, North-Holland, Amsterdam, 1973, pp. 98–100. MR 0360800
Zvonimir Janko and John G. Thompson, On a class of finite simple groups of Ree, J. Algebra 4 (1966), 274–292. MR 201504, DOI 10.1016/0021-8693(66)90041-X
Lary Schiefelbusch, Sylow $2$-subgroups of simple groups, J. Algebra 31 (1974), 131–153. MR 357599, DOI 10.1016/0021-8693(74)90010-6

Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen

132

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 104733