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Author: Ross Lawther
Publisher: American Mathematical Soc.
Published: 1999
Total Pages: 146
ISBN-13: 0821819666
DOWNLOAD EBOOKThis book is intended for graduate students and research mathematicians interested in group theory and genralizations
Author: Adam R. Thomas
Publisher: American Mathematical Soc.
Published: 2021-06-18
Total Pages: 191
ISBN-13: 1470443376
DOWNLOAD EBOOKThis paper is a contribution to the study of the subgroup structure of excep-tional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group G is called irreducible if it lies in no proper parabolic subgroup of G. In this paper we com-plete the classification of irreducible connected subgroups of exceptional algebraic groups, providing an explicit set of representatives for the conjugacy classes of such subgroups. Many consequences of this classification are also given. These include results concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of irreducible connected subgroups are determined by their composition factors on the adjoint module of G, with one exception. A result of Liebeck and Testerman shows that each irreducible connected sub-group X of G has only finitely many overgroups and hence the overgroups of X form a lattice. We provide tables that give representatives of each conjugacy class of connected overgroups within this lattice structure. We use this to prove results concerning the subgroup structure of G: for example, when the characteristic is 2, there exists a maximal connected subgroup of G containing a conjugate of every irreducible subgroup A1 of G.
Author: Martin W. Liebeck
Publisher: American Mathematical Soc.
Published: 2012-01-25
Total Pages: 394
ISBN-13: 0821869205
DOWNLOAD EBOOKThis 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.
Author: Martin W. Liebeck
Publisher: American Mathematical Soc.
Published: 1996
Total Pages: 122
ISBN-13: 0821804618
DOWNLOAD EBOOKThe theory of simple algebraic groups is important in many areas of mathematics. The authors of this book investigate the subgroups of certain types of simple algebraic groups and obtain a complete description of all those subgroups which are themselves simple. This description is particularly useful in understanding centralizers of subgroups and restrictions of representations.
Author: Alastair J. Litterick
Publisher: American Mathematical Soc.
Published: 2018-05-29
Total Pages: 168
ISBN-13: 1470428377
DOWNLOAD EBOOKThe study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic.
Author: Martin W. Liebeck
Publisher: American Mathematical Soc.
Published: 2004
Total Pages: 242
ISBN-13: 0821834827
DOWNLOAD EBOOKIntends to complete the determination of the maximal subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields. This title follows work of Dynkin, who solved the problem in characteristic zero, and Seitz who did likewise over fields whose characteristic is not too small.
Author: Gunter Malle
Publisher: Cambridge University Press
Published: 2011-09-08
Total Pages: 324
ISBN-13: 113949953X
DOWNLOAD EBOOKOriginating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.
Author: Gary M. Seitz
Publisher: American Mathematical Soc.
Published: 1987
Total Pages: 294
ISBN-13: 0821824279
DOWNLOAD EBOOKLet [italic]V be a finite dimensional vector space over an algebraically closed field of characteristic p [greater than] 0 and let G = SL([italic]V), Sp([italic]V), or SO([italic]V). The main result describes all closed, connected, overgroups of [italic]X in SL([italic]V), assuming [italic]X is a closed, connected, irreducible subgroup of G.
Author: David A. Craven
Publisher: American Mathematical Society
Published: 2022-04-08
Total Pages: 168
ISBN-13: 1470451190
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Author: Ross Lawther
Publisher: American Mathematical Soc.
Published: 1999-09-01
Total Pages: 148
ISBN-13: 9780821863978
DOWNLOAD EBOOKAbstract. Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p$. Under some mild restrictions on $p$, we classify all conjugacy classes of closed connected subgroups $X$ of type $A_1$; for each such class of subgroups, we also determine the connected centralizer and the composition factors in the action on the Lie algebra ${\mathcal L}(G)$ of $G$. Moreover, we show that ${\mathcal L}(C_G(X))=C_{{\mathcal L}(G)}(X)$ for each subgroup $X$. These results build upon recent work of Liebeck and Seitz, who have provided similar detailed information for closed connected subgroups of rank at least $2$. In addition, for any such subgroup $X$ we identify the unipotent class ${\mathcal C}$ meeting it. Liebeck and Seitz proved that the labelled diagram of $X$, obtained by considering the weights in the action of a maximal torus of $X$ on ${\mathcal L}(G)$, determines the ($\mathrm{Aut}\,G$)-conjugacy class of $X$. We show that in almost all cases the labelled diagram of the class ${\mathcal C}$ may easily be obtained from that of $X$; furthermore, if ${\mathcal C}$ is a conjugacy class of elements of order $p$, we establish the existence of a subgroup $X$ meeting ${\mathcal C}$ and having the same labelled diagram as ${\mathcal C}$.
