J Austral Math Soc Ser A 49 pp502--513, 1990.

Centralizing Properties in Simple Locally Finite Groups and Large Classical Groups

Brian Hartley

Abstract

The following question is discussed and evidence for and against it is advanced: is it true that if F is an arbitrary finite subgroup of an arbitrary non-linear simple locally finite group G, then C_G(F) is infinite? The following points to an affirmative answer. THEOREM A. Let F be an arbitrary finite subgroup of a non-linear simple locally finite group G. Then there exists subgroups DÑC £ G such that F centralizes C/D, FÇC £ D, and C/D is a direct product of finite alternating groups of unbounded orders. In particular, F centralizes an infinite section of G. Theorem A is deduced from a "local" version, namely THEOREM B. There exists an integer valued function f(n, r) with the following properties. Let H be a finite group of order at most n, and suppose that H £ S, where S is either an alternating group of degree at least f = f(n, r) or a finite classical group whose natural projective representation has degree at least f. Then there exist subgroups DÑC £ S such that (i) [H, C] £ D, (ii) H ÇC £ D, (iii) C/D @ Alt(r), (iv) D = 1 if S is alternating, and D is a p-group of class at most 2 and exponent dividing p² if S is a classical group over a field of characteristic p. The natural "local version" of our main question is however definitely false. PROPOSITION C. Let p br a given prime. Then there exists a finite group H that can be embedded in infinitely many groups PSL(n, p) as a subgroup with trivial centralizer.

1980 AMS Subject Classification (1985 Revision): 20F50, 20D08

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Authors

Brian Hartley

University of Manchester, Manchester M13 9PL, England.