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On the Fixity of Linear Groups
Author(s) -
Shalev Aner
Publication year - 1994
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-68.2.265
Subject(s) - mathematics , automorphism , combinatorics , group (periodic table) , dimension (graph theory) , pure mathematics , bounded function , normal subgroup , extension (predicate logic) , mathematical analysis , chemistry , organic chemistry , computer science , programming language
The fixity of a linear group G ⊆ GL( V ) is defined to be the maximal dimension of a centraliser C v ( g ) for g ∈ G with g ≠ 1. We study the structure of finite non‐modular linear groups of given fixity f . Our results may be regarded as an extension of the theory of Frobenius complements on the one hand, and of Jordan's classical theorem on finite subgroups of GL n (C) on the other. As an application we show that if G is a group of automorphisms of a finite group H , and each non‐trivial element of G fixes at most f points of H , then G has a soluble subgroup of derived length at most 3 whose index is bounded above in terms of f alone.