Applications of Group Theory to Combinatorics pdf epub mobi txt 电子书 下载 2026

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出版者:CRC Press

作者:Koolen, Jack (EDT)/ Kwak, Jin Ho (EDT)/ Xu, Ming-yao (EDT)

出品人:

页数:188

译者:

出版时间:2008-06-01

价格:USD 144.95

装帧:Hardcover

isbn号码:9780415471848

丛书系列:

图书标签:

• Group Theory
• Combinatorics
• Algebra
• Mathematics
• Permutation Groups
• Enumerative Combinatorics
• Graph Theory
• Symmetry
• Mathematical Structures
• Abstract Algebra

Applications of Group Theory to Combinatorics: A Deep Dive into Abstract Structures and Discrete Worlds This volume explores the profound and often unexpected intersections between the abstract machinery of group theory and the tangible world of combinatorial enumeration and structure. Rather than presenting a survey of existing applications, this book focuses on developing a cohesive framework that leverages group-theoretic concepts—such as symmetry, permutation representation, and character theory—to solve challenging problems in counting, design theory, coding theory, and graph enumeration. The central theme is the translation of combinatorial objects into representations of specific groups, allowing powerful algebraic tools to illuminate otherwise intractable counting problems. The initial chapters lay the groundwork by establishing the necessary algebraic foundations without assuming prior advanced knowledge of representation theory. We begin with a rigorous treatment of permutation groups, focusing intently on the concept of transitivity and the structure of stabilizers. This is crucial, as most combinatorial applications stem from analyzing the action of a group on a set of configurations (e.g., colorings, labelings, or arrangements). We introduce the Orbit-Stabilizer Theorem not just as a counting tool, but as the fundamental principle underpinning the Pólya Enumeration Theorem (PET). The heart of the book lies in the comprehensive treatment of the Pólya Enumeration Theorem (PET) and its generalizations. We move beyond simple cycle index calculations, dedicating substantial sections to the application of PET in analyzing patterns under various symmetry groups. This includes detailed case studies on enumerating necklaces, bracelets, and polyominoes, meticulously deriving the cycle indices for dihedral, cyclic, and full symmetry groups relevant to geometric arrangements. Crucially, we extend the discussion to Exponential Generating Functions (EGFs) within the context of PET, exploring how exponential structures (like rooted trees or functional mappings) necessitate the use of EGFs instead of ordinary generating functions (OGFs), providing a complete picture of how structural constraints influence the appropriate generating function framework. A significant departure from standard texts is the dedicated focus on Burnside's Lemma as a pre-cursor to Character Theory in Enumeration. While Burnside's Lemma often suffices for basic counting under permutation groups, we demonstrate its limitations when dealing with weighted enumeration or structures where the coloring scheme itself imposes algebraic constraints. This motivates the introduction of character theory. We systematically develop the concepts of irreducible representations, characters, and orthogonality relations for finite groups. The connection is forged by showing how the structure of the permutation representation of the group action on the set of colorings can be decomposed into irreducible components. This decomposition provides a far more nuanced count than simple fixed-point counting, particularly useful in advanced topics like enumerating inequivalent labelings of graphs or molecules. The second half of the book shifts focus to more specialized combinatorial domains where group theory plays a structural, rather than purely enumerative, role. Group Theory in Design Theory and Coding: We explore how the concept of automorphism groups defines the equivalence classes of combinatorial designs (such as Steiner systems or block designs). We analyze the search for specific symmetric designs (e.g., Hadamard matrices or projective planes) by treating the incidence structure as a set acted upon by a presumed group of symmetries. The constraint that the structure must be invariant under the group action significantly prunes the search space. Following this, we introduce coding theory, specifically focusing on Group Codes and Cyclic Codes. Here, the group structure is intrinsic to the code construction itself—often leveraging the cyclic group $mathbb{Z}_n$ or related finite fields. We demonstrate how the algebraic properties derived from Fourier transforms over these groups (akin to character theory) lead directly to efficient methods for calculating minimum distances and correcting errors. Symmetry and Graph Theory: A major section is devoted to the enumeration of non-isomorphic graphs, rooted and unrooted, under the action of various automorphism groups. We detail techniques for constructing adjacency matrices and analyzing their spectral properties in relation to the group action. The application of PET to map colorings is expanded to encompass labeling problems on graphs, providing explicit formulas for counting labelings that respect certain symmetries. Furthermore, we delve into the construction of Cayley graphs and vertex-transitive graphs, where the very definition of the graph relies upon the algebraic structure of the underlying group and a chosen set of generators. Analyzing the properties of these graphs—such as connectivity and diameter—becomes inherently linked to the group's internal structure. Advanced Applications: Quantum Information and Algebra: For advanced readers, the final chapters touch upon cutting-edge applications. We explore how tensor products of permutation representations lead to tools applicable in the nascent field of quantum combinatorics, particularly in analyzing non-local correlations via group-theoretic constraints (e.g., in Bell inequalities derived from group actions). Furthermore, we provide a detailed study on the use of Sylow Theorems not for enumeration, but for guaranteeing the existence of specific subgroups within the automorphism groups of highly regular combinatorial structures, which can simplify the structural analysis dramatically. Throughout the text, emphasis is placed on concrete examples rooted in geometry and finite sets, ensuring that the transition from abstract group axioms to practical counting formulas is transparent and robust. The book aims not merely to show that group theory works, but how its foundational concepts of action, invariance, and representation structure provide the essential language for modern combinatorial analysis.

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这本书《群论在组合学中的应用》给我最大的感受是，它成功地将一个相对抽象的数学分支，转化成了一种解决问题的“利器”。书中大量的例子，涵盖了从基础的排列组合问题，到更复杂的图论和编码理论中的应用，都展示了群论的强大威力。我特别欣赏作者在介绍“Burnside's Lemma”时，并非简单地给出一个公式，而是通过一系列生动形象的例子，比如计算具有不同颜色的项链，让你深刻理解这个引理的核心思想。这本书让我认识到，原来很多我们看似困难的计数问题，都可以通过理解其背后的对称性，并通过群论的工具来优雅地解决。书中并没有回避一些相对复杂的数学概念，但作者的处理方式非常到位，往往会先给出一个直观的解释，然后再逐步深入到数学的细节。这对于我这种希望能够深入理解问题本质的读者来说，是极大的帮助。我甚至觉得，这本书的价值并不仅限于其本身的内容，它更是一种思维模式的塑造，让我学会如何用更宏观、更系统化的视角去分析和解决问题。

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坦白讲，我之前对群论和组合学都只有一些零散的了解，感觉它们是两个独立且各自独立的领域。《群论在组合学中的应用》这本书，却以一种令人惊喜的方式，将这两个领域紧密地联系在了一起。它并没有刻意地去强调数学的严谨性，而是更侧重于群论在解决组合学问题时的“巧思”。我特别喜欢书中关于“Polya Enumeration Theorem”的介绍，它不仅仅是提供了一个计数公式，更重要的是揭示了如何通过分析对象的对称群来设计有效的计数策略。书中对于不同类型的对称性，例如旋转对称、反射对称等，都有非常详尽的讨论，并且将其与组合学问题巧妙地结合起来。我甚至觉得，这本书的意义远不止于组合学领域，它能够培养读者一种“结构化思考”的能力，让你在面对任何看似混乱的问题时，都能尝试去寻找其内在的结构和对称性，从而找到解决问题的突破口。这本书的语言风格非常平实，没有过多的修饰，但字里<bos>. of 确切，如同工匠般打磨出来的精品。

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这本《群论在组合学中的应用》真是让人眼前一亮，尤其是对于我这种初涉组合学领域的读者而言。它不像我之前看过的那些教科书，上来就抛出一堆抽象的概念和复杂的公式，而是用一种非常生动、循序渐进的方式，将群论这个看似“高深莫测”的数学工具，巧妙地融入到了各种有趣的组合学问题之中。我尤其喜欢书中关于“伯恩赛德引理”和“波利亚计数定理”的讲解，它们不仅仅是理论推导，更是通过大量的实例，比如数数不同的项链排列、不同的骰子染面等等，将抽象的计数问题变得直观易懂。我甚至可以想象自己带着这本书，在咖啡馆里，一边品着咖啡，一边跟着作者的思路，一点点地解开那些看似无解的谜题，那种感觉真是妙不可言。书中还涉及到了一些图论中的应用，比如判断图的同构性，这对于我理解一些算法的本质非常有帮助。虽然有些地方的数学符号我还需要花点时间去消化，但整体而言，这本书给我打开了一扇新的大门，让我看到了数学的优雅与力量，也激发了我进一步探索组合学和群论的兴趣。我迫不及待地想把我学到的东西应用到我自己的研究项目中，相信这本书一定会成为我案头常备的参考书。

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说实话，拿到《群论在组合学中的应用》这本书，我最初的预期是它会像市面上很多同类书籍一样，充斥着繁复的证明和枯燥的定义。然而，这本书完全超出了我的想象。它并非一股脑儿地堆砌理论，而是以一种非常“接地气”的方式，将群论的抽象概念与组合学中那些我们熟悉的、甚至是有些“顽固”的问题一一对应。我印象最深刻的是关于“对称性”的处理。在组合学中，很多计数问题都源于对象的对称性，而群论恰恰是研究对称性的强大工具。书中通过生动的例子，比如计算特定形状的棋盘上不同颜色的着色方式，是如何利用群的结构来消除重复计数，让我豁然开朗。我甚至觉得，这本书更像是一本“应用指南”，它不只是告诉你“是什么”，更重要的是告诉你“怎么用”。它的逻辑清晰，每一章的过渡都非常自然，仿佛作者是在和我这个读者进行一场深入的对话，循循善诱地引导我一步步理解其中的精髓。虽然我目前的专业方向并非数学，但这本书的数学语言并不生涩，很多地方都巧妙地用图示和类比来辅助理解，这对于非数学专业的读者来说，简直是福音。

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这本书《群论在组合学中的应用》的价值，我认为主要体现在它对抽象概念的“具象化”处理。很多时候，我们学习数学工具，往往停留在理论层面，却难以将其真正运用到解决实际问题上。这本书的出现，则很好地弥补了这一断层。它通过引入一系列经典的组合学问题，例如如何计算不同化学分子结构的同分异构体数量，或者是在设计某些编码系统时如何利用群论来保证其鲁棒性，让我们看到了群论在现实世界中的强大生命力。书中对“轨道-稳定子定理”的讲解，就让我对如何通过对称性来简化计数问题有了全新的认识。我尤其赞赏作者在处理复杂问题时，能够将其拆解成更小的、更容易理解的部分，并通过反复的示例来加深读者的理解。有时候，我会暂停下来，尝试自己去解决书中提出的某些小挑战，然后对照作者的解答，这种互动式的学习体验，让我受益匪浅。这本书不仅仅是知识的传递，更是一种思维方式的启迪，它让我学会了如何从对称性和结构的角度去审视和解决问题，这对于我未来的学习和工作都将产生深远的影响。

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