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14.46 (1999)

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A finite group $G$ is said to be almost simple if $T \leqslant G \leqslant \operatorname{Aut}(T)$ for some nonabelian simple group $T$. By definition, a finite linear space consists of a set $V$ of points, together with a collection of $k$-element subsets of $V$, called lines ($k \geqslant 3$), such that every pair of points is contained in exactly one line. Classify the finite linear spaces which admit a line-transitive almost simple subgroup $G$ of automorphisms which acts transitively on points.

A. Camina, P. Neumann and C. E. Praeger have solved this problem in the case where $T$ is an alternating group. In (A. Camina, C. E. Praeger, Aequat. Math., 61 (2001), 221–232) it is shown that a line-transitive group of automorphisms of a finite linear space which is point-quasiprimitive (i. e. all of whose non-trivial normal subgroups are point-transitive) is almost simple or affine.

Contributor: A. Camina, C. E. Praeger

Progress

Remarks of 2005: The cases of the following groups, but not almost simple groups with this socle, were dealt with: $PSU(3, q)$ (W. Liu, Linear Algebra Appl., 374 (2003), 291–305); $PSL(2, q)$ (W. Liu, J. Combin. Theory (A), 103 (2003), 209–222); $Sz(q)$ (W. Liu, Discrete Math., 269 (2003), 181–190); $Ree(q)$ (W. Liu, Europ. J. Combin., 25 (2004), 311–325); sporadic simple groups were done in (A. R. Camina, F. Spiezia, J. Combin. Des., 8 (2000), 353–362). Remark of 2009: It was shown (N. Gill, Trans. Amer. Math. Soc., 368 (2016), 3017–3057) that for non-desarguesian projective planes a point-quasiprimitive group would be affine. Remark of 2013: The problem is solved for large-dimensional classical groups (A. Camina, N. Gill, A. E. Zalesski, Bull. Belg. Math. Soc. Simon Stevin, 15, no. 4 (2008), 705–731).

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