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.
Publié le : 2008-05-15
Classification:
linear space,
block design,
line-transitive,
finite classical group,
05B05,
20B25,
20D06
@article{1225893950,
author = {Camina, Alan R. and Gill, Nick and Zalesski, A.E.},
title = {Large dimensional classical groups and linear spaces},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {15},
number = {1},
year = {2008},
pages = { 705-731},
language = {en},
url = {http://dml.mathdoc.fr/item/1225893950}
}
Camina, Alan R.; Gill, Nick; Zalesski, A.E. Large dimensional classical groups and linear spaces. Bull. Belg. Math. Soc. Simon Stevin, Tome 15 (2008) no. 1, pp. 705-731. http://gdmltest.u-ga.fr/item/1225893950/