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Partition backtrack is the current generic state of the art algorithm to search for subgroups of a given permutation group. We describe an improvement of partition backtrack for set stabilizers and intersections of subgroups by using orbital graphs. With extensive experiments we demonstrate that our methods improve performance of partition backtrack -in some cases by several orders of magnitude.

In this article we classify all L 9 -free finite groups.

Abstract.In this paper we analyze the structure of transitive permutation groups that have trivial four point stabilizers, but some nontrivial three point stabilizer. In particular, we give a complete, detailed classification when the group is simple or quasisimple. This paper is motivated by questions concerning the relationship between fixed points of automorphisms of Riemann surfaces and Weierstraß points and is a continuation of the authors' earlier work.

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