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"This book presents the solution of a long-standing problem concerning the stable module category (of not necessarily finite dimensional representations) of a finite group. The proof draws on commutative algebra, cohomology of groups and stable homotopy theory. The unifying theme is a notion of support which provides a geometric approach for studying various algebraic structures. "

"Dalam teori grup, terdapat teorema yang menyatakan bahwa suatu grup abelian berhingga dapat didekomposisi menjadi hasil kali langsung dari subgrup-subgrupnya. Karena order dari grup abelian berhingga dapat difaktorkan menjadi perkalian dari bilangan-bilangan prima, maka subgrup hasil dekomposisinya adalah grup siklik. Salah satu penggunaan dari hasil dekomposisi ini adalah menghitung order automorfisma grup abelian berhingga melalui perhitungan order automorfisma dari grup-grup hasil dekomposisinya. Dalam skripsi ini dibahas mengenai pembuktian dekomposisi grup abelian berhingga menjadi hasil kali langsung dari grup-grup siklik dan perhitungan order automorfisma dari grup abelian berhingga.

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In group theory, there is a theorem which states that a finite abelian group can be decomposed into a direct product of its subgroups. Since an order of the finite abelian group can be factored into product of prime numbers,then the resulting decomposition subgroups are cyclic groups. One of these applications of the theory is calculating the authomorphism order of a finite abelian group through calculation of automorphism order from the group as its decomposition result.This mini thesis discussed about the proof of finite abelian group into direct product of its cyclic groups and the calculation of its automorphism order from the finite abelian group."

"Starting with matrices over the integers, part 1 of this book provides a measured introduction to such groups, two finitely generated abelian groups are isomorphic if and only if their invariant factor sequences are identical. The analogous theory of matrix similarity over a field is then developed in part 2 starting with matrices having polynomial entries, two matrices over a field are similar if and only if their rational canonical forms are equal. Under certain conditions each matrix is similar to a diagonal or nearly diagonal matrix, namely its Jordan form."

"These lecture notes provide a tutorial review of non-Abelian discrete groups and show some applications to issues in physics where discrete symmetries constitute an important principle for model building in particle physics. While Abelian discrete symmetries are often imposed in order to control couplings for particle physics, in particular model building beyond the standard model - non-Abelian discrete symmetries have been applied to understand the three-generation flavor structure in particular. This text explicitly introduces and studies the group-theoretical aspects of many concrete groups and shows how to derive conjugacy classes, characters, representations, and tensor products for these groups (with a finite number) when algebraic relations are given, thereby enabling readers to apply this to other groups of interest."

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Lema Lima Pendek adalah sebuah pernyataan mengenai kategori Abelian. Untuk mendefinisikan kategori Abelian, perlu dibahas mengenai kategori. Teori kategori adalah sebuah teori yang memberikan konteks untuk menyatukan berbagai macam struktur matematis. Suatu kategori terdiri atas sebuah koleksi berisi objek beserta himpunan-himpunan berisi morfisma sedemikian sehingga beberapa aksioma terpenuhi. Sebuah kategori dapat digambarkan sebagai sebuah diagram berpanah dengan setiap panah dipandang sebagai sebuah morfisma. Pada teori kategori, beberapa konsep seperti produk dan kernel dapat didefinisikan. Kategori Abelian merupakan sebuah kategori yang diberikan strukturstruktur Abelian serta memenuhi beberapa aksioma tambahan. Pada kategori Abelian, dapat didefinisikan konsep barisan eksak pendek. Lema Lima Pendek merupakan sebuah hasil sederhana mengenai barisan eksak pendek. Pada tugas akhir ini, diberikan bukti alternatif Lema Lima Pendek.

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The Short Five Lemma is a statement about Abelian categories. To define Abelian categories, the notion of categories needs to be discussed. Category theory is a theory which gives a context for combining various mathematical structures. A category consists of a collection of objects and sets of morphisms such that a set of axioms are satisfied. A category can be drawn as a diagram consisting of arrows where an arrow is considered as a morphism. In category theory, concepts such as products and kernels are introduced. An Abelian category is a category equipped with Abelian structures which satisfies some additional axioms. In Abelian categories, the notion of a short exact sequence is defined. The Short Five Lemma is a simple result regarding short exact sequences. In this thesis, an alternative proof of the Short Five Lemma is presented.

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