Infinite Groups: A Roadmap to Selected Classical Areas

ISBN-13: 9780367702625 / Angielski / Twarda / 2023 / 406 str.

Igor Ya. Subbotin

Opis

This book covers a broad swath of the theory of infinite groups, without giving proofs, but with all the concepts and auxiliary results necessary for understanding such results. In other words, this book is an extended directory, or a guide, to some of the more established areas of infinite groups.

 

Spis treści

1. Important Subgroups. 1.1. Some Important Series in Groups and Subgroups Defined by these Series. 1.2. Classes of Groups Defined by Series of Subgroups. 1.3. Radicable Groups. 1.4. Something from the Theory of Modules. 1.5. The 0-Rank and p-rank of Abelian Groups. 1.6. The Frattini Subgroup of a Group. 1.7. Linear Groups. 1.8. Residually X-Groups. 2. Finitely Generated Groups. 2.1. The Generalized Burnside Problem. 2.2. The Burnside Problem for Groups of Finite Exponent. 2.3. The Restricted Burnside Problem. 2.4. Growth Functions on Finitely Generated Groups. 2.5. Finitely Presented Groups. 2.6. Groups with the Maximal Condition for all Subgroups. 3. Finiteness Conditions. 3.1 The Minimal Condition on Certain Systems of Subgroups. 3.2. The Minimal Condition on Normal Subgroups. 3.3. Artinian and Related Modules over some Group Rings. 3.4. Minimax Groups. 3.5. The Weak Minimal Condition. 3.6. The Weak Maximal Condition. 4. Ranks of Groups. 4.1. Finite Special Rank and Finite Section p-Rank. 4.2. Finite 0-Rank. 4.3. The Connections Between the Various Rank Conditions I. 4.4. Finite Section Rank. 4.5. Bounded Section Rank. 4.6. The Connections Between the Various Rank Conditions II. 4.7. Finitely Generated Groups. 4.8. Systems of Subgroups Satisfying Rank Conditions. 4.9. Some Residual Systems. 5. Conjugacy Classes. 5.1. Around "Schur’s Theorem", Central-By-Finite Groups and Related Topics. 5.2 Bounded Conjugacy Classes, Finite-By-Abelian Groups and Related Classes. 5.3. Groups with Finite Classes of Conjugate Elements. 5.4. Some Concluding Remarks. 6. Generalized Normal Subgroups and their Opposites. 6.1. Groups Whose Subgroups are Normal, Permutable or Subnormal. 6.2. Groups having a Large Family of Normal Subgroups. 6.3. Groups having a Large Family of Subnormal Subgroups. 6.4. Pairs of Opposite Subgroups. 6.5. Transitively Normal Subgroups. 6.6. The Norm of a Group, The Wielandt Subgroup and Related Topics. 6.7. The Norm of a Group and the Quasicentralizer Condition. 7. Locally Finite Groups. 7.1. Preliminaries. 7.2. Large Locally Finite Groups. 7.3. Simple Locally Finite Groups. 7.4. Existentially Closed Groups. 7.5. Centralizers in Locally Finite Groups. 7.6. Sylow Theory in Locally Finite Groups. 7.7. Conjugacy of Sylow Subgroups. 7.8. Unconventional Sylow Theories. 7.9. Saturated Formations and Fitting Classes. 7.10. Barely Transitive Groups.