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"Fundamentals of group theory provides a comprehensive account of the basic theory of groups. Both classic and unique topics in the field are covered, such as an historical look at how Galois viewed groups, a discussion of commutator and Sylow subgroups, and a presentation of Birkhoff’s theorem. "

"Groups are a means of classification, via the group action on a set, but also the object of a classification. How many groups of a given type are there, and how can they be described? Hölder’s program for attacking this problem in the case of finite groups is a sort of leitmotiv throughout the text. Infinite groups are also considered, with particular attention to logical and decision problems. Abelian, nilpotent and solvable groups are studied both in the finite and infinite case. Permutation groups and are treated in detail; their relationship with Galois theory is often taken into account. The last two chapters deal with the representation theory of finite group and the cohomology theory of groups, the latter with special emphasis on the extension problem. The sections are followed by exercises; hints to the solution are given, and for most of them a complete solution is provided."

"Teori kategori adalah teori yang mendalami abstraksi morfisma atau pemetaan antar struktur matematika. Suatu kategori terdiri dari objek dan morfisma serta memenuhi dua aksioma yaitu aksioma asosiatif dan aksioma identitas. Kategori bertujuan untuk membangun konsep fungtor dan transformasi alami. Fungtor merupakan pengaitan antar kategori dan transformasi alami merupakan pengaitan antar fungtor. Dari fungtor dan transformasi alami, dapat dibangun sebuah konsep ekuivalensi yang menjelaskan ’kesamaan’ suatu struktur kategori. Kategori monoidal merupakan kategori dengan tambahan sifat monoid, yaitu memiliki operasi biner berupa bifungtor, asosiator berupa isomorfisma alami, dan objek unit berupa objek 1 beserta dua unitor yang merupakan isomorfisma alami. Kategori monoidal memenuhi dua aksioma, yaitu aksioma segilima dan aksioma segitiga. Bila objek yang dikaitkan oleh asosiator dan unitor adalah sama, maka diperoleh sifat ketegasan (strictness). Kategori monoidal dengan sifat ketegasan (strictness) disebut sebagai kategori monoidal tegas (strict). Teorema Ketegasan Mac Lane menyatakan bahwa setiap kategori monoidal ekuivalen monoidal dengan suatu kategori monoidal tegas. Penulisan skripsi ini bertujuan untuk mengkaji dan menuliskan kembali bukti Teorema Ketegasan Mac Lane pada kategori monoidal.

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Category theory is a theory that explores the abstraction of morphisms or the mapping between mathematical structures. A category consists of objects and morphisms and satisfies two axioms, namely the associative axiom and the axiom of identity. Categories aim to build the concept of functors and natural transformations. A functor is a mapping between categories and a natural transformation is a mapping between functors. From functors and natural transformations, an equivalence concept can be constructed that explains the ’similarity’ of categories. Monoidal categories are categories with the addition of monoidal properties, namely having a binary operation in the form of a bifunctor, an associator in the form of a natural isomorphism, and a unit object in the form of object 1 and two unitors which are natural isomorphisms. A monoidal category satisfies two axioms, namely the pentagon axiom and the triangular axiom. If the object associated by the associator and unitor is the same, then a new characteristic appears, namely strictness. A monoidal category with strictness is referred to as a strict monoidal category. Mac Lane’s Strictness Theorem states that every monoidal category is monoidally equivalent to a strict monoidal category. The writer aims to examine and rewrite the proof of Mac Lane’s Strictness Theorem on monoidal categories."

"Spectral radius of graphs provides a thorough overview of important results on the spectral radius of adjacency matrix of graphs that have appeared in the literature in the preceding ten years, most of them with proofs, and including some previously unpublished results of the author. The primer begins with a brief classical review, in order to provide the reader with a foundation for the subsequent chapters. Topics covered include spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem. From this introduction, the book delves deeper into the properties of the principal eigenvector; a critical subject as many of the results on the spectral radius of graphs rely on the properties of the principal eigenvector for their proofs. A following chapter surveys spectral radius of special graphs, covering multipartite graphs, non-regular graphs, planar graphs, threshold graphs, and others. Finally, the work explores results on the structure of graphs having extreme spectral radius in classes of graphs defined by fixing the value of a particular, integer-valued graph invariant, such as: the diameter, the radius, the domination number, the matching number, the clique number, the independence number, the chromatic number or the sequence of vertex degrees."

"This book is about modern algebraic geometry. The book starts by explaining this enigmatic answer, algebraic geometry. From a point of departure in algebraic curves, the exposition moves on to the present shape of the field, culminating with Alexander Grothendieck’s theory of schemes. Contemporary homological tools are explained."