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出版者:

作者:Segal, Daniel

出品人:

页数:304

译者:

出版时间:2005-11

价格:$ 80.23

装帧:

isbn号码:9780521023948

丛书系列:

图书标签:

• 群论
• 多循环群
• 代数拓扑
• 抽象代数
• 数学
• 李群
• 有限群
• 代数结构
• 群表示论
• 同调代数

Polycyclic Groups: A Journey into the Structure of Infinite Groups This book offers a comprehensive exploration of polycyclic groups, a class of infinite groups that possess remarkable structural properties akin to those of finite soluble groups. Far from being a mere catalog of theorems, "Polycyclic Groups" delves into the very heart of these structures, illuminating their construction, classification, and the intricate relationships they hold with other fundamental concepts in group theory and beyond. The narrative begins by grounding the reader in the foundational concepts essential for understanding polycyclic groups. We will systematically build the theoretical framework, starting with the indispensable notion of a subnormal series and its crucial role in dissecting group structure. The book meticulously defines and explains the properties of soluble groups, establishing their significance as a stepping stone towards grasping the nuances of polycyclic groups. Crucially, we will introduce and elaborate on the concept of an FC-group, a class of groups where every conjugacy class is finite, and demonstrate how this property often underlies the behavior of polycyclic groups. The interplay between these foundational ideas sets the stage for the central subject. The core of the book is dedicated to the precise definition and characterization of polycyclic groups. We will introduce the concept of a sequence of subgroups, each normal in the next, with cyclic factors. This definition, while elegant, hints at a rich and complex structure. The book will then systematically prove the fundamental theorem that characterizes polycyclic groups: a group is polycyclic if and only if it has a finite subnormal series with cyclic factors. This theorem is not just a technical result; it is the key that unlocks the understanding of their finite-like behavior. We will explore various equivalent characterizations and demonstrate how these seemingly different definitions coalesce to reveal the same underlying structure. A significant portion of the book is dedicated to the constructive aspects of polycyclic groups. We will explore how these groups can be built from simpler components, focusing on the concept of direct products and semidirect products. The book will meticulously illustrate how combining cyclic groups in specific ways can lead to the formation of polycyclic groups. Furthermore, we will delve into the crucial concept of nilpotency and solubility within the context of polycyclic groups, demonstrating how these properties are intimately linked and how they can be exploited to simplify the analysis of their structure. The book will showcase the relationship between polycyclic groups and their derived series, providing a powerful tool for understanding their internal organization and identifying their soluble radical. Classification is a cornerstone of abstract algebra, and this book provides a thorough treatment of the classification of polycyclic groups. We will explore the concept of maximal normal subgroups and their role in constructing a maximal chain of normal subgroups, leading to a canonical representation of polycyclic groups. The book will introduce the notion of the Fitting subgroup and its significance in understanding the nilpotency structure of polycyclic groups. We will also investigate the relationship between polycyclic groups and their automorphism groups, a topic that reveals deep insights into the symmetries and structural rigidity of these groups. The book will present a detailed discussion of the generators and relations for polycyclic groups, providing a concrete way to describe and manipulate these abstract objects. Beyond their intrinsic structural beauty, polycyclic groups find significant applications and connections to other areas of mathematics. The book will explore their role in algebraic geometry, particularly in the study of algebraic groups. We will demonstrate how polycyclic groups arise naturally as subgroups of these larger structures and how their properties can illuminate the behavior of the entire algebraic group. Furthermore, the book will touch upon their relevance in number theory, especially in the context of Galois theory and the study of solvable extensions of fields. The book will also explore their connections to representation theory, illustrating how the properties of polycyclic groups influence the representations they admit. Throughout "Polycyclic Groups," emphasis is placed on rigorous proofs, clear explanations, and a wealth of examples. The book is designed for a graduate student or researcher in algebra who wishes to gain a deep and nuanced understanding of this important class of groups. We aim to equip the reader with the theoretical tools and conceptual insights necessary to tackle problems involving polycyclic groups and to appreciate their fundamental role in the landscape of infinite group theory. The journey through polycyclic groups is one of discovery, revealing a world of intricate structure, elegant classifications, and surprising connections that continue to fascinate mathematicians.

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说实话，这本书的印刷质量和装帧设计，透着一股浓厚的学术气息，让人一上手就觉得这不是一本消遣读物。我尤其欣赏作者在阐述非交换群的结构时所采用的视角。不同于一些侧重于有限群的教材，这本书非常注重无限群，特别是那些具有复杂正规子群结构的群的描述。它对挠群（Torsion Groups）和自由群的讨论，简直是教科书级别的典范。作者并没有将这些复杂的概念碎片化处理，而是始终保持着宏观的视角，让读者能够清晰地看到，当群的阶趋于无穷大时，其内部的复杂性是如何以一种既有序又迷人的方式爆发出来的。书中对群作用的几何解释部分，虽然篇幅不算多，但却如同沙漠中的绿洲，为那些沉浸在符号运算中的读者提供了一个急需的直观锚点。我个人认为，这本书的价值在于它不仅仅是知识的传递，更是一种思维方式的培养——它训练你如何在一个高度结构化的体系内进行严谨的、多层次的思考，这对于任何从事理论研究的人来说，都是一笔宝贵的财富。

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我得承认，阅读这本关于“多环群”的专业书籍，对我来说是一次真正的智力挑战，但回报也是巨大的。这本书的叙述风格非常古典，可以说是对传统代数教科书美学的极致体现。它大量依赖于纯粹的逻辑推导和定理的完美证明链条。如果你习惯了现代教材中那种插图丰富、讲解口语化的风格，那么初读此书可能会感到有些吃力。它的行文节奏相对缓慢而审慎，每一个定理的提出都像是经过深思熟虑的，并且其证明过程详尽到令人敬佩。特别是关于范畴论在群论中的应用那几章，作者的处理方式极其优雅，他没有简单地罗列定义，而是通过一系列精心构造的函子，展示了不同群结构之间的“桥梁”。不过，我必须提醒初学者，这本书的难度曲线相当陡峭。如果你只是想快速了解一些皮毛，它可能不是最佳选择。但如果你渴望深入到这个领域的“骨骼”层面，理解那些看似坚不可摧的结构是如何被逻辑的铁锤精心打磨出来的，那么这本书就是一座无可替代的灯塔。它要求读者保持高度的专注力，但相信我，当你最终掌握了其中一个关键概念时，那种掌握真理的满足感是无可比拟的。

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这本关于多环群的专著，读起来不像是在看一本工具书，倒更像是在阅读一份关于宇宙基本规律的哲学宣言。作者的写作风格非常个人化，带着一种近乎诗意的数学语言。他对于“简单群”的引入和探讨，简直是大师级的处理。他没有直接罗列那些上千个简单群的分类结果，而是通过一系列思想实验，引导读者去思考，究竟是什么样的结构能够达到这种“不可再分解”的状态。书中对群的表示论的侧重，也让我耳目一新。相比于其他教材侧重于纯粹的群论本体，这本书巧妙地将线性代数的力量引入进来，使得抽象的群操作在矩阵的变换中获得了具象的表达。我喜欢这种跨学科的对话，它让原本可能枯燥的理论变得鲜活起来。不过，也正因为这种风格的独特性，这本书对读者的预备知识要求极高，如果对基础的抽象代数概念掌握不牢固，很可能会在开篇几章就迷失方向。总而言之，这是一本需要慢品、反复咀嚼才能体会到其深意的杰作，它不是为了快速解答问题而写的，而是为了启发更深刻的探究而存在的。

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我不得不说，在众多群论书籍中，这本书在处理群的分解和组合性质方面，展现出了非凡的洞察力。它对斯里（Sylow）定理的讨论，简直是教科书上应该采用的范本。作者不仅给出了标准的证明，还深入探讨了这些定理背后的代数几何意义，以及它们在判定有限群结构时的普适性。更让我印象深刻的是关于半直积分解的应用，书中详尽地分析了如何利用这种分解来构造和识别那些看似不规则的群。这种自下而上、层层递进的构建方法，极大地增强了读者对群结构清晰的认知。当然，本书的内容密度非常高，每一页都塞满了深刻的数学信息，阅读时必须保持警惕，生怕错过任何一个关键的脚注或引文。它不是那种让你轻松翻阅的读物，更像是一块需要用耐心和时间去雕琢的璞玉。对于希望在有限群的结构理论上打下坚实基础的研究者来说，这本书提供的深度和广度是其他书籍难以匹及的，它真正做到了将复杂性管理得井井有条，让读者在敬畏中学习。

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啊，这本关于多环群的著作，我算是终于啃完了。说实话，刚拿到手的时候，我就被它那厚重的分量和封面那种略带古朴的字体给震慑住了。这本书的结构安排得非常精妙，它没有一上来就抛出那些让人望而生畏的抽象定义，而是像一位经验丰富的向导，带着我们一步步深入。作者显然对这个领域有着深刻的理解，从最基础的循环群和有限阿贝尔群讲起，慢慢过渡到更复杂的结构。尤其值得称道的是，书中对于群的生成元、同态映射以及子群的讨论，都配有大量的例子，这些例子并非那种教科书式的简单重复，而是巧妙地揭示了不同概念之间的内在联系。我记得有一次，我对着一个复杂的商群结构冥思苦想了很久，翻到书中的某个例子后，突然间茅塞顿开，那种豁然开朗的感觉，至今记忆犹新。这本书的数学严谨性毋庸置疑，但更重要的是，它成功地将抽象的代数概念“人格化”了，让你感觉这些群的结构就像一个个有生命的实体在相互作用，而不是冰冷的符号堆砌。对于那些希望系统性学习群论，特别是对超越基础内容有追求的读者来说，这本书绝对是案头必备的工具书。

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