Bulletin of the Belgian Mathematical Society - Simon Stevin

Large dimensional classical groups and linear spaces

Alan R. Camina, Nick Gill, and A.E. Zalesski

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Abstract

Suppose that a group $G$ has socle $L$ a simple large-rank classical group. Suppose furthermore that $G$ acts transitively on the set of lines of a linear space $\mathcal{S}$. We prove that, provided $L$ has dimension at least $25$, then $G$ acts transitively on the set of flags of $\mathcal{S}$ and hence the action is known. For particular families of classical groups our results hold for dimension smaller than $25$. The group theoretic methods used to prove the result (described in Section 3) are robust and general and are likely to have wider application in the study of almost simple groups acting on finite linear spaces.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin Volume 15, Number 4 (2008), 705-731.

Dates
First available in Project Euclid: 5 November 2008

Permanent link to this document
http://projecteuclid.org/euclid.bbms/1225893950

Mathematical Reviews number (MathSciNet)
MR2475494

Zentralblatt MATH identifier
05496972

Subjects
Primary: 05B05: Block designs [See also 51E05, 62K10] 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX] 20D06: Simple groups: alternating groups and groups of Lie type [See also 20Gxx]

Keywords
linear space block design line-transitive finite classical group

Citation

Camina, Alan R.; Gill, Nick; Zalesski, A.E. Large dimensional classical groups and linear spaces. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 4, 705--731. http://projecteuclid.org/euclid.bbms/1225893950.


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