## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### Large dimensional classical groups and linear spaces

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

#### 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. Bulletin of the Belgian Mathematical Society - Simon Stevin 15 (2008), no. 4, 705--731. http://projecteuclid.org/euclid.bbms/1225893950.