Normalizers of Primitive Permutation Groups

Let $G$ be a transitive normal subgroup of a permutation group $A$ of finite degree $n$. The factor group $A/G$ can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that $|A/G| < n$ if $G$ is primitive unless $n = 3^{4}$, $5^4$, $3^8$, $5^8$, or $3^{16}$. This bound is sharp when $n$ is prime. In fact, when $G$ is primitive, $|\mathrm{Out}(G)| < n$ unless $G$ is a member of a given infinite sequence of primitive groups and $n$ is different from the previously listed integers. Many other results of this flavor are established not only for permutation groups but also for linear groups and Galois groups.Comment: 44 pages, grant numbers updated, referee's comments include

Simple groups admit Beauville structures

We answer a conjecture of Bauer, Catanese and Grunewald showing that all finite simple groups other than the alternating group of degree 5 admit unmixed Beauville structures. We also consider an analog of the result for simple algebraic groups which depends on some upper bounds for character values of regular semisimple elements in finite groups of Lie type and obtain definitive results about the variety of triples in semisimple regular classes with product 1. Finally, we prove that any finite simple group contains two conjugacy classes C,D such that any pair of elements in C x D generates the group.Comment: 30 pages, in the second version, some results are improved and in particular we prove an irreducibility for a certain variet

Cocompact lattices on A~_n buildings

We construct cocompact lattices Γ’&lt;sub&gt;0&lt;/sub&gt;&lt; Γ&lt;sub&gt;0&lt;/sub&gt; in the group G = PGL&lt;sub&gt;d&lt;/sub&gt;(F&lt;sub&gt;q&lt;/sub&gt;((t))) which are type-preserving and act transitively on the set of vertices of each type in the building Δ associated to G. These lattices are commensurable with the lattices of Cartwright [Steger [CS]. The stabiliser of each vertex in Γ’&lt;sub&gt;0&lt;/sub&gt; is a Singer cycle and the stabiliser of each vertex in Γ&lt;sub&gt;0&lt;/sub&gt; is isomorphic to the normaliser of a Singer cycle in PGL&lt;sub&gt;d&lt;/sub&gt;(q). We show that the intersections of Γ’&lt;sub&gt;0&lt;/sub&gt; and Γ&lt;sub&gt;0&lt;/sub&gt; with PSL&lt;sub&gt;d&lt;/sub&gt;(F&lt;sub&gt;q&lt;/sub&gt;((t))) are lattices in PSL&lt;sub&gt;d&lt;/sub&gt;(F&lt;sub&gt;q&lt;/sub&gt;((t))), and identify the pairs (d; q) such that the entire lattice Γ’&lt;sub&gt;0&lt;/sub&gt; or Γ&lt;sub&gt;0&lt;/sub&gt; is contained in PSL&lt;sub&gt;d&lt;/sub&gt;(F&lt;sub&gt;q&lt;/sub&gt;((t))). Finally we discuss minimality of covolumes of cocompact lattices in SL&lt;sub&gt;3&lt;/sub&gt;(F&lt;sub&gt;q&lt;/sub&gt;((t))). Our proofs combine the construction of Cartwright{Steger [CS] with results about Singer cycles and their normalisers, and geometric arguments