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Primitive block designs with automorphism group \(\operatorname{PSL}(2,q)\). (English) Zbl 1360.05021

The paper considers primitive block designs with the projective line \(F_q\cup\{\infty \}\) as the set of points and \(\operatorname{PSL}(2,q)\) as a predefined automorphism group, where \(q=p^n\) and \(p\) is prime. A \(t\)-design is primitive if it is invariant under an automorphism group which acts primitively on the point and block sets.
The orbit of a \(k\)-subset of a point set on which a \(t\)-homogeneous group acts can be viewed as a \(t\)-design. The authors obtain primitive 2- and 3-designs from the \(k\)-subsets (\(k\leq q/2\)) whose set-wise stabilizer is a maximal subgroup of the predefined automorphism group. They use known facts about the subgroup structure of finite classical groups and plenty of combinatorial reasoning to determine the existence conditions, parameters, isomorphism and full automorphism groups of the resultant designs.
For the designs with block stabilizers in the fifth Aschbacher’s class the classification is for \(q=p\), while in the other cases \(q=p^n\), \(p\)-prime. Some of the obtained 3-designs have been considered in previous works, but the isomorphism and automorphism results about them are new.

MSC:

05B05 Combinatorial aspects of block designs

Software:

Magma; GAP; DESIGN
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