European Mathematical Society - i. m. isaacs
https://euro-math-soc.eu/author/i-m-isaacs
enAlgebra. A Graduate Course
https://euro-math-soc.eu/review/algebra-graduate-course
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This textbook introduces the reader to methods of modern algebra. It is based on a graduate algebra course but the author's aim is more ambitious - the book attempts to present both results and methods of abstract algebra as an exciting and beautiful part of modern mathematics. The fact that this edition is reprinted by the American Mathematical Society from the original that appeared in 1994, which became one of the most popular textbooks on algebra, reflects the fact that the author's aim has met with obvious success. The book is divided into two parts. The first of them is devoted to noncommutative algebra (following the author's professional interests and his conception of organisation of algebraic topics). The first ten chapters, almost the whole first third of the book, build group theory. The rest of the noncommutative part contains an introduction to the theory of modules and non-commutative rings. Elements of character theory, which forms a natural link between module theory and the theory of groups, are presented in the last chapter of the first part of the book. The commutative algebra part opens with a chapter devoted to polynomial rings, principal ideal rings and unique factorisation domains and is followed by a nine-chapter section covering field theory. The last five chapters present classical topics from commutative algebra and algebraic geometry, including primary decompositions of commutative noetherian rings, Dedekind domains and the Nullstellensatz. </p>
<p>The book covers almost all standard algebraic topics (except homological and categorical algebra) and, moreover, it includes several advanced parts of group theory (e.g. transfer theory). Note that the textbook is almost self-contained since a reader only needs an elementary algebraic background (elements of linear algebra and the basic concept of mathematical structures). Last but not least, a feature of the book that should be mentioned is the number of carefully chosen problems, which are listed at the end of every chapter. The textbook is excellent and an unusually successful combination of the best of traditional introduction to algebra, for which accuracy and correctness are absolutely essential, and a modern text that can be recommended to graduate students of algebra and all those whose interests lie in pure mathematics as well as everyone who wishes or needs to become familiar with the beauty of modern algebra.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jž</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/i-m-isaacs" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">i. m. isaacs</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-mathematical-society" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american mathematical society</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2009</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-8218-4799-2 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 63</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/12-field-theory-and-polynomials" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">12 Field theory and polynomials</a></li></ul></span>Tue, 07 Jun 2011 22:12:55 +0000Anonymous39381 at https://euro-math-soc.euFinite Group Theory
https://euro-math-soc.eu/review/finite-group-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>There are many textbooks on group theory. This one is aimed at graduate students who know the basics (however, there is an appendix that covers the introductory topics) and who seek solid knowledge of classical techniques. The book is written with obvious care and the exposition is clear and covers many topics that are not very accessible or not very well explained elsewhere. For example, chapter 1 is essentially about Sylow theorems and consequences but it also covers the Chermak-Delgado measure. Chapter 2 is on subnormality (the Wielandt zipper lemma and the theorems of Baer, Zenkov and Lucchini). Chapter 3 presents the standard material on split extensions (e.g. Hall subgroups and Glaubermann's lemma). Chapter 4, called Commutators, contains, amongst other things, various consequences of the Hall-Witt identity, the theorem of Mann and Thompson's PxQ lemma. </p>
<p>Chapter 5 introduces the technique of transfer, digresses into infinite groups by proving theorems of Schur and Dietzmann, continues with the Burnside theorem on normal p-complements, introduces focal subgroups and finishes with the Frobenius theorem on normal p-complements. The nilpotency of Frobenius kernel is proved in chapter 6. In fact, a part of the proof relies upon the properties of the Thompson subgroup, to which chapter 7 is devoted. This chapter also presents a proof of Burnside's theorem that is independent of linear representation theory (the proof follows arguments of Goldschmidt, Matsuyama and Bender). The ensuing chapter contains standard material on permutation groups, followed by a discussion of the orbital graph and subdegrees, including the proofs of the theorems of Weiss and Manning. The penultimate chapter starts with a definition of the generalised Fitting subgroup and of components and continues with a discussion of their basic properties. It also proves Wielandt's results on automorphism towers, Schenkman’s theorem, Thompson's result on corefree maximal subgroups (in connection to the Sims conjecture) and some facts about strong conjugacy. The final chapter returns to the transfer (theorems of Yoshida, Huppert's result on metacyclic groups and connections with the group ring). Each chapter has several subsections (A, B, C, D and sometimes E, F and G) and each subsection finishes with a list of problems, many of which are far from routine.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">ad</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/i-m-isaacs" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">i. m. isaacs</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-mathematical-society" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american mathematical society</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2008</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-8218-4344-4 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 59</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/20-group-theory-and-generalizations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">20 Group theory and generalizations</a></li></ul></span>Mon, 30 May 2011 18:54:36 +0000Anonymous39255 at https://euro-math-soc.eu