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Finite simple groups. --- Geometric group theory. --- Maximal subgroups.

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This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.

Finite groups. --- Maximal subgroups. --- Fitting subgroup --- Subgroup, Fitting --- Subgroups, Maximal --- Group theory --- Groups, Finite --- Modules (Algebra) --- Finite groups --- Maximal subgroups --- Mathematical models.

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"This monograph is a contribution to the study of the subgroup structure of exceptional 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 complete 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 subgroup 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"--

Linear algebraic groups. --- Representations of groups. --- Embeddings (Mathematics) --- Maximal subgroups. --- Groupes algébriques linéaires --- Représentations de groupes --- Plongements (mathématiques)

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Group theory --- Frattini subgroups. --- Conjugacy classes. --- Embeddings (Mathematics) --- Plongements (mathématiques) --- Conjugacy classes --- Frattini subgroups --- Subgroups, Frattini --- Maximal subgroups --- Imbeddings (Mathematics) --- Geometry, Algebraic --- Immersions (Mathematics) --- Classes of conjugate elements

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Linear algebraic groups --- Maximal subgroups --- Representations of groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Fitting subgroup --- Subgroup, Fitting --- Subgroups, Maximal --- Algebraic groups, Linear --- Geometry, Algebraic --- Algebraic varieties --- Linear algebraic groups. --- Maximal subgroups. --- Representations of groups. --- Représentations de groupes --- Groupes algébriques linéaires --- Représentations de groupes. --- Groupes algébriques linéaires.