This book is a clear and compelling presentation of the fundamental concepts and notions in the area of group ranks. Class-tested worldwide by highly qualified authors in the fields of abstract algebra and group theory, the book focuses on carefully selected group-theoretic notions, namely the various ranks of a group and classical numerical functions related to groups. The existence of these functions related to groups makes it possible to achieve significant progress in the study and clarification of the structure of groups. In order to provide the necessary tools, characterizations, and restrictions of ranks in a group, the authors have chosen the most up-to-date topics in the field of group theory from hundreds of current research articles. Featuring an efficient presentation of the various ranks, the book focuses on the central concepts with the most interesting, striking, and central results. The authors prove these results using various methods to illustrate the role these ranks have in the resulting definitions. Covering the various topics in a logical manner, the authors begin with a succinct presentation of the standard ranks before moving on to more in depth aspects of ranks of groups. The topical coverage includes: Section ranks; groups of finite 0-rank; minimax rank; special rank of groups; groups of finite section p-rank; structure of groups having finite section p-rank for all primes p; groups of finite bounded section rank; groups whose abelian subgroups have finite ranks; groups whose abelian subgroups have bounded finite ranks; finitely generated groups having finite rank; residual properties of groups of finite rank; groups covered by normal subgroups of bounded finite rank; and Schur and Baer theorems.

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