Finding characters satisfying a maximal condition for their unipotent support

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor indecomposable -restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V ) of this form, where H is a closed disconnected almost simple positive-dimensional subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classifcation of the irreducible triples (G,H,V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension.

FIELD AUTOMORPHISMS AND MAXIMAL SUBGROUPS OF FINITE CHEVALLEY GROUPS

We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type. Introduction Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F ) and positive integers n . The topic of this survey is motivated by the question as to what happens if GL ( n ,·) is replaced by another algebraic group G (·). We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k . A further restriction of our discussion concerns the field k . It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G (ℂ). (See below for some exceptions in §3 and §5.) For G (·) of classical type, the theory for representations L → G (ℂ) differs little from the usual one for GL ( n , ℂ). Indeed, a representation L → GL ( n , ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The ‘trichotomy’ here is a case division of the generic minimal counterexamples within odd type, that is, groups with a large and divisible Sylow° 2-subgroup. The so-called ‘uniqueness case’ in the trichotomy theorem is the existence of a proper 2-generated core. It is our aim to drive the presence of a proper 2-generated core to a contradiction, and hence bind the complexity of the Sylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilber conjecture. This paper shows that the group in question is a minimal connected simple group and has a strongly embedded subgroup, a far stronger uniqueness case. As a corollary, a tame counterexample to the Cherlin–Zilber conjecture has Prüfer rank at most two.

The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H1, …, Hm of some special form such that π(H1)∪…∪π(Hm) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections Hi so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H1,…,Hm} controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control.

We show that if H is a reductive group, then nth roots of conjugacy classes are a finite union of conjugacy classes, and that if G is an algebraic overgroup of H, then the intersection of H with a conjugacy class of G is a finite union of H-conjugacy classes. These results follow from results on finiteness of unipotent classes in an almost simple algebraic group.

A signalizer functor theorem for groups of finite Morley rank

A ‐tuple of core‐free subgroups of a finite group is said to be regular if has a regular orbit on the Cartesian product . The regularity number of , denoted by , is the smallest positive integer with the property that every such ‐tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple groups with an alternating or sporadic socle. For example, we prove that and . We also formulate and investigate natural generalisations of several well‐studied problems on base sizes for finite permutation groups, including conjectures due to Cameron, Pyber and Vdovin. For instance, we extend earlier work of Burness, O'Brien and Wilson by proving that for every almost simple sporadic group, with equality if and only if is the Mathieu group . We also show that every triple of soluble subgroups in an almost simple sporadic group is regular, which generalises recent work of Burness on base sizes for transitive actions of sporadic groups with soluble point stabilisers.

Simple classical groups of Lie type are determined by their character degrees

On Engel words on simple algebraic groups

Let be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐ modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient . For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

A theorem of Cassidy states that any Zariski closed differential algebraic subgroup of a simple linear algebraic group, defined over a differential field, is either the whole group or is conjugate to the subgroup of constant matrices. An arithmetic analogue of this theorem is proved in which usual derivations on fields are replaced by certain nonlinear operators on p -adic rings, called " p -derivations."

Chapter 1. A classification of Curtis-Tits amalgams.- Chapter 2. The use of valuations for classifying point-line geometries.- Chapter 3. An outline of polar spaces: basics and advances.- Chapter 4. Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties.- Chapter 5. Generation of Lie Incidence Geometries: A Survey.- Chapter 6. Witt-Type Theorems for Subspaces of Lie Geometries: A Survey.- Chapter 7. Embeddings of cotriangular spaces.- Chapter 8. Unipotent Overgroups in Simple Algebraic Groups.- Chapter 9. The axes of a Majorana representation of A12.- Chapter 10. GIT related problems of the flag variety for the action of a maximal torus.- Chapter 11. Characterizations of trialities of type Iid in buildings of type D4.- Chapter 12. On the isomorphism problem for Coxeter groups and related topics.- Chapter 13. Lectures on Artin groups and the K(pi 1) conjecture. Chapter 14. Algebraic codes and geometry of some classical generalized polygons.- Chapter 15. Some Weyl modules of the algebraic groups of type E_6.- Problem Set.

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

By a classical theorem of Jordan, every faithful transitive action of a non‐trivial finite group has a derangement (an element with no fixed points). The existence of derangements with additional properties has attracted much attention, especially for faithful primitive actions of almost simple groups. In this paper, we show that an almost simple group can have an element that is a derangement in every faithful primitive action, and we call these elements totally deranged. In fact, we classify the totally deranged elements of all almost simple groups, showing that an almost simple group contains a totally deranged element only if the socle of is or with . Using this, we classify the invariable generating sets of a finite simple group of the form where and , answering a question of Garzoni. As a final application, we classify the elements of almost simple groups that are contained in a unique maximal subgroup in the case where is not core‐free, which complements the recent work of Guralnick and Tracey addressing the case where is core‐free.