Transitive permutation groups with trivial four point stabilizers

Abstract

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.

Finite simple groups acting with fixity 3 and their occurrence as groups of automorphisms of Riemann surfaces (extended version)

“…Remark For the proof of the theorem, we recall some relevant definitions, then we refer to [8] for the specific groups to consider and analyse them with the methods developed in [11] and by using [2].…”

“…Proof. By hypothesis G acts with fixity 3, so we may apply Theorem 1.1 in [8] with the additional information that the point stabilisers are cyclic to obtain the result.…”

“…We combine the results from [7] and [8] with the additional hypothesis that the point stabilisers are cyclic. More specifically we inspect Theorem 1.1 in [8] and Lemmas 3.11 and 3.13 of [7]. This gives exactly the groups that are mentioned in the lemma.…”

“…The first input for the list [PSL 2 (7), g, 0 | [3, 2], [7,1]], together with its output, shows that we need to enter the element orders in the tuple "orders" of the function GenTuple1 with multiplicities! gap> GenTuple1(AlternatingGroup( 7 (3,8,6,4), (2,4,6,5,8,3,7) ] gap> GenTuple1(PSL(2,7),0, [3,4,4,7]); Computation complete : 2 orbits found. 42 tuples in orbit 1 42 tuples in orbit 2 Picking random tuple in random orbit ...…”

“…In parallel to [11], we now build on the classification results in [8], for fixity 3, with the aim to give information about the curves such that the finite simple groups from [8] act as groups of automorphisms and in such a way that every non-trivial element fixes at most three points on each orbit and at most four points in total. We describe the curves and the action of the group in terms of branching data, which does not give an explicit isomorphism type of the curve.…”