On the fixity of linear groups

Aner Shalev*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


The fixity of a linear group G GL(V) is defined to be the maximal dimension of a centraliser Cv(g) for 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 GLn(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.

Original languageAmerican English
Pages (from-to)265-293
Number of pages29
JournalProceedings of the London Mathematical Society
Issue number2
StatePublished - Mar 1994

Bibliographical note

Funding Information:
The author is grateful to the Mathematics Research Section of the Australian National University for its kind hospitality while this work was carried out. 1991 Mathematics Subject Classification: 20D99, 20G15.


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