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This book provides a complete and comprehensive classification of normal 2-coverings of non-abelian simple groups and their generalizations. While offering readers a thorough understanding of these structures, and of the groups admitting them, it delves into the properties of weak normal coverings. The focal point is the weak normal covering number of a group G, the minimum number of proper subgroups required for every element of G to have a conjugate within one of these subgroups, via an element of Aut(G). This number is shown to be at least 2 for every non-abelian simple group and the non-abelian simple groups for which this minimum value is attained are classified. The discussion then moves to almost simple groups, with some insights into their weak normal covering numbers. Applications span algebraic number theory, combinatorics, Galois theory, and beyond. Compiling existing material and synthesizing it into a cohesive framework, the book gives a complete overview of this fundamental aspect of finite group theory. It will serve as a valuable resource for researchers and graduate students working on non-abelian simple groups,
List of contents
- Introduction.- Preliminaries.- Linear groups.- Unitary groups.- Symplectic groups.- Odd dimensional orthogonal groups.- Orthogonal groups with Witt defect 1.- Orthogonal groups with Witt defect 0.- Proofs of the main theorems.- Almost simple groups having socle a sporadic simple group.- Dropping the maximality.- Degenerate normal 2-coverings.
About the author
Daniela Bubboloni is Professor of Mathematics at the University of Florence (Italy). Her main research interests involve finite groups and graph theory. A peculiar aspect of her research is the application of permutation groups and graph theory to questions of social choice theory, a branch of economics.
Pablo Spiga is Professor of Mathematics at the University of Milano-Bicocca (Italy). His main research interests involve group actions on graphs and other combinatorial structures. His main expertise is in finite primitive groups and their application to the investigation of symmetries of combinatorial structures.
Thomas Weigel is Professor of Mathematics at the University of Milano-Bicocca (Italy). His research interests are in the theory of groups, their representation theory and their cohomology theory. His main expertise is in the theory of groups of Lie-type, and the structure theory and cohomology of profinite groups.
Report
The purpose of this monograph is the classification of finite non-abelian simple groups ... . the authors give various motivations for their investigation. Besides the intrinsic theoretical interest in (weak) normal 2-coverings, a few remarkable applications have arisen ... . Each of these applications has a number of interesting open questions and conjectures ... . The reviewer believes that the presence of a subject index ... would have greatly aided the scholar in reading this interesting work. (Enrico Jabara, zbMATH 1548.20003, 2024)
