Gary M. Seitz

On the minimal degrees of projective representations of the finite Chevalley groups

For G = G(q), a Chevalley group defined over the field iFQ of characteristic p, let Z(G,p) be th e smallest integer t > 1 such that G has a projective irreducible representation of degree t over a field of characteristic other than p. In this paper we present lower bounds for the numbers Z(G,p). As a corollary we determine those Chevalley groups having an irreducible complex character of prime degree. Recently there have been a number of results making use of lower bounds on the degrees of representations of Chevalley groups. See for example Curtis, Kantor, and Seitz [4], Hering [9], and Patton [l 11. Also in Fong and Seitz [7] such bounds played an important role, although there the representationsconsideredwere over fields of characteristicp. For most types of Chevalley groups and for most primes p it is not difficult to obtain reasonable lower bounds for the complex irreducible characters of G = G(q), using the existence of certain p-subgroups of G resembling extraspecial groups. Indeed this was carried out in Landazuri [lo]. However to be complete we must take into certain problems that occur with fields of characteristic 2 and 3. Also, since we are considering projective irreducible representations the groups with exceptional Schur multipliers present some dif3iculties. There is also the problem of deciding whether or not a lower bound is “good.” In some cases our bounds are actually attained and there is no problem in this regard. Otherwise let {G(q)} be a family of Chevalley groups of given type and with q ranging over suitable prime powers. Then our bounds will be in the form of a polynomial in q. In Curtis, Iwahori, and Kilmoyer [3] there is a list of certain character degrees for the family (G(p))

The 2-transitive permutation representations of the finite Chevalley groups

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Finite groups with a split BN-pair of rank 1. II

In Part I of this paper (Hering, Kantor, and Seitz [SO]), 2-transitive groups of even degree were classified when the stabilizer of a point has a normal subgroup regular on the remaining points. The identification with groups of known type was made by finding a 2-Sylow subgroup and then applying the deep classification theorems of Alperin, Brauer and Gorenstein [l, 21 and Walter [39]. The purpose of the present continuation of [50] is to point out that the proof of the main result of [50] can be completed without using [I] and [2]. 1Ioreover, Walter’s classification theorem [39] and the Gorenstein-Walter Theorem [49] are not required in [50], although the end of Walter [53] seems to be needed. Our arguments are natural continuations of those of [50, Sections 4, 8, and 91. Much use is also made of character-theoretic information contained in Brauer [46] and [47]. Our goal is to show that a minimal counteresamplc has a cyclic two points stabilizer G,,, and then apply a result of Kantor, O’Nan and Seitz [22, Theorem 1.1 or Section 5, Case D]. Re first show that G,, is metacyclic, and then “transfer out field automorphisms” in order to prove that G,, is cyclic. This transfer argument yielded an unexpected dividend: in the course of examining a similar argument in Suzuki [34, Section 211, an error was found. This has been corrected, and, in fact, the entire transfer argument is stated for odd and even degree groups simultaneously. ‘Phc numbering of both the sections and the references will be continued from [50].

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers.

On the subgroup structure of exceptional groups of Lie type

We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X(q) of Lie type in the natural characteristic. Our approach is to show that for sufficiently large q (usually q > 9 suffices), X(q) is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.

Some representations of exceptional Lie algebras

In this note, we give the dimensions of some irreducible representations of exceptional Lie algebras and algebraic groups. Similar results appear in [1] for classical groups and algebras of rank at most 4. These results were produced by computer programs developed in connection with [3], where the main result required information beyond the tables in [1]. In view of the utility of the tables in [1], it seemed worthwhile to provide tables for groups of higher rank. Although our methods are similar to those of [l], they incorporate a reduction process which permits us to push the techniques a bit further.

Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable

We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let D be a finite dimensional division algebra having center K and let N ⊆ D× be a normal subgroup of finite index. Suppose D×/N is not solvable. Then we may assume that H := D×/N is a minimal nonsolvable group (MNS group for short), i.e., a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote property (3 12 ). This property includes the requirement that the diameter of the commuting graph should be ≥ 3, but is, in fact, stronger. Another ingredient is to show that if the commuting graph of D×/N has the property (312 ), then N is open with respect to a nontrivial height one valuation of D (assuming without loss, as we may, that K is finitely generated). After establishing the openness of N (when D×/N is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of K over its prime subfield, to eliminate H as a possible quotient of D×, thereby obtaining a contradiction and proving our main result.

Finite and Locally Finite Groups

Preface. Introduction. Simple locally finite groups B. Hartley. Algebraic groups G.M. Seitz. Subgroups of simple algebraic groups and related finite and locally finite groups of Lie type M.W. Liebeck. Finite simple groups and permutation groups J. Saxl. Finitary linear groups: a survey R.E. Phillips. Locally finite simple groups of finitary linear transformations J.I. Hall. Non-finitary locally finite simple groups U. Meierfrankenfeld. Inert subgroups in simple locally finite groups V.V. Belyaev. Group rings of simple locally finite groups A.E. Zalesskii. Simple locally finite groups of finite Morley rank and odd type A.V. Borovik. Existentially closed groups in specific classes F. Leinen. Groups acting on polynomial algebras R.M. Bryant. Characters and sets of primes for solvable groups I.M. Isaacs. Character theory and length problems A. Turull. Finite p-groups A. Shalev. Index.

Subgroups of Algebraic Groups Containing Regular Unipotent Elements

Let G be a simple algebraic group over an algebraically closed field K of characteristic p with p [ges ]0. Then G contains finitely many conjugacy classes of unipotent elements. There is a unique class of unipotent elements of largest dimension – the class of regular unipotent elements ; the centralizer of any such element is a unipotent group of dimension equal to the rank of G . For example, if G is the linear group SL ( V ) on the finite dimensional vector space V over K , then the regular unipotent elements are precisely the unipotent matrices consisting of a single Jordan block.

Subgroups of finite groups of Lie type

Let G :G(p) be a finite group of Lie type defined over the field F, . Choose a Bore1 subgroup, B = UH . 11. Associated with G is a root system, z, and a collection of root subgroups {U, : a: E Z> such that C’ 1 nI,,,Zrz and such that FI < N(U,) foreach a E ,X In Lemma 3 of [ 1 I] it was shown that for q :‘-4 any H-invariant subgroup of U is essentially a product of root subgroups (the word “essentially” is relevant only when G is twisted, with some root subgroup non-Abelian). This I-es& was cxtcnded in [7], where it was shown that any unipotent subgroup of G normalized by H is of this form (although now negative roots are allowed).