M. C. Tamburini

2-Generation of finite simple groups and some related topics

It is well-known that every finite simple group is 2-generated, i.e. it can be generated by two suitable elements. This is the topic discussed in §1, which centers around Steinberg’s unified treatment of groups of Lie type. In §2 we discuss generation of simple groups by special kinds of generating pairs, namely: 1) the generation of simple groups of Lie type by a cyclic maximal torus and a long root element, with application to the solution of the Magnus-Gorchakov-Levchuk conjecture on residual properties of free groups; 2) the generation of a simple group by an involution and another suitable element. With regard to 1), we also mention similar 2-generation results in connection with Galois groups; with regard to 2), emphasis is put on (2,3)-generation and Hurwitz generation of finite simple groups. Finally, §3 deals with generating sets of involutions of minimal size. Most finite simple groups are generated by three involutions. Generation results, a non-generation criterion, and a relation between (2,3)-generation and generation by three involutions are illustrated.

HURWITZ GROUPS OF LARGE RANK

A finite non-trivial group G is called a Hurwitz group if it is an image of the infinite triangle group formula here Thus G is a Hurwitz group if and only if it can be generated by an involution and an element of order 3 whose product has order 7. The history of Hurwitz groups dates back to 1879, when Klein [ 9 ] was studying the quartic formula here of genus 3. The automorphism group of this curve has order 168 = 84(3−1), and it is isomorphic to the simple group PSL 2 (7), which is generated by the projective images of the matrices formula here with product formula here and so is a Hurwitz group. In 1893, Hurwitz [ 7 ] proved that the automorphism group of an algebraic curve of genus g (or, equivalently, of a compact Riemann surface of genus g ) always has order at most 84( g −1), and that, moreover, a finite group of order 84( g −1) can act faithfully on a curve of genus g if and only if it is an image of Δ(2, 3, 7). The problem of determining which finite simple groups are Hurwitz groups has received considerable attention. In [ 10 ], Macbeath classified the Hurwitz groups of type PSL 2 ( q ); there are infinitely many of them. In [ 1 ] Cohen proved that no group PSL 3 ( q ) is a Hurwitz group except PSL 3 (2), which is isomorphic to PSL 2 (7). Certain exceptional groups of Lie type, and some of the sporadic groups, are known to be Hurwitz groups. For discussions of the results on these groups we refer the reader to [ 3 , 5 , 11 ].

ON (2, 3, 7)-GENERATION OF MAXIMAL PARABOLIC SUBGROUPS

Let R be a ring with 1 and £„(/?) be the subgroup of GLn(R) generated by the matrices / + retj, r e R, i ^ j. We prove that the subgroup Pnn of En+a (R) consisting of the matrices of shape ( A o j) , where A € £„(/?). A € Ej,(R) and B e MatnS(/?), is (2, 3, 7)-generated whenever R is finitely generated and n, h are large enough.

Tensor products of primitive modules

Abstract. Let F be a field and, for i = 1,2, let Gi be a group and Vi an irreducible, primitive, finite dimensional FGi-module. Set G = G1\times G2 and

Classical Groups of Large Rank as Hurwitz Groups

V=V_1\otimes _F V_2

On hurwitz groups of low rank

. The main aim of this paper is to determine sufficient conditions for V to be primitive as a G-module. In particular this turns out to be the case if V1 and V2 are absolutely irreducible and V1 is absolutely quasi-primitive. Thus we extend a result of N.S. Heckster, who has shown that V is primitive whenever G is finite and F is the complex field. We also give a characterization of absolutely quasi-primitive modules. Ultimately, our results rely on the classification of finite simple groups.