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Line-transitive point-imprimitive linear spaces with number of points being a product of two primes. (English) Zbl 1364.05016

Summary: The work studies the line-transitive point-imprimitive automorphism groups of finite linear spaces, and is underway on the situation when the numbers of points are products of two primes. Let \(\mathcal{S}\) be a non-trivial finite linear space with \(cd\) points, where \(c\) and \(d\) are two primes. We prove that if \(G\leq \operatorname{Aut}(\mathcal{S})\) is line-transitive point-imprimitive, then \(G\) is solvable.

MSC:

05B05 Combinatorial aspects of block designs
05B25 Combinatorial aspects of finite geometries
20B15 Primitive groups
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures

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