Composition Series In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ..., a composition series provides a way to break up an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ..., such as a group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ... or a module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and en ... [...More Info...]       [...Related Items...] Simple Group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a simple group is a nontrivial group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ... whose only normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...s are the trivial groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathem ... [...More Info...]       [...Related Items...] Noetherian Module In abstract algebra, a Noetherian module is a module (mathematics), module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion (set theory), inclusion. Historically, David Hilbert, Hilbert was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated module, finitely generated. However, the property is named after Emmy Noether who was the first one to discover the true importance of the property. Characterizations and properties In the presence of the axiom of choice,{{fact, date=November 2016 two other characterizations are possible: *Any nonempty set ''S'' of submodules of the module has a maximal element (with respect to set inclusion.) This is known as the maximum condition. *All of the submodules of the module are ... [...More Info...]       [...Related Items...] picture info Fundamental Theorem Of Arithmetic In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ..., a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the fundamental theorem of arithmetic, also called the unique factorization theorem, states that every integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ... greater than 1 can be represented uniquely as a product of prime number A ... [...More Info...]       [...Related Items...] picture info Cyclic Group In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ..., a branch of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ..., a cyclic group or monogenous group is a group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ... that is generated by a single element. That is, it is a set of invertible elements with a single associative In mathematics Math ... [...More Info...]       [...Related Items...] picture info Transfinite Induction Transfinite induction is an extension of mathematical induction Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement ''P''(''n'') holds for every natural number ''n'' = 0, 1, 2, 3, . . . ; that is, the overall statement is a ... to well-ordered sets, for example to sets of ordinal number In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...s or cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...s. Induction by cases Let P(\alpha) be a property Property is a system of rights that ... [...More Info...]       [...Related Items...] Schreier Refinement Theorem In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there is a bijection between their factor groups that sends each factor group to an group isomorphism, isomorphic one. The theorem is named after the Austrian mathematician Otto Schreier who proved it in 1928. It provides an elegant proof of the Jordan–Hölder theorem. It is often proved using the Zassenhaus lemma. gives a short proof by intersecting the terms in one subnormal series with those in the other series. Example Consider \mathbb/(2) \times S_3, where S_3 is the symmetric group of degree 3. The alternating group A_3 is a normal subgroup of S_3, so we have the two subnormal series : \ \times \ \; \triangleleft \; \mathbb/(2) \times \ \; \triangleleft \; \mathbb/(2) \times S_3, : \ \times \ \; \triangleleft \; \ \times A_3 \; \triangleleft \; \mathbb/(2) \times S_3, with res ... [...More Info...]       [...Related Items...] Isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., an isomorphism is a structure-preserving mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ... between two structures A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A sy ... of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. ... [...More Info...]       [...Related Items...] picture info Permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from s, which are selections of some members of a set regardless of order. For example, written as s, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. s of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of s is an important topic in the fields of and . Permutations are used in almost every branch of mathematics, and in many other fields of science. In , they are used for analyzing s; in , for describing states of particles; and in , fo ... [...More Info...]       [...Related Items...] picture info Up To Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ... objects ''a'' and ''b'' are called equal up to an equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... of ''a'' and ''b'' with respec ... [...More Info...]       [...Related Items...] Otto Hölder Otto Ludwig Hölder (December 22, 1859 – August 29, 1937) was a Germany, German mathematician born in Stuttgart. Hölder first studied at the ''Polytechnikum'' (which today is the University of Stuttgart) and then in 1877 went to Berlin where he was a student of Leopold Kronecker, Karl Weierstrass, and Ernst Kummer. He is noted for many theorems including: Hölder's inequality, the Jordan–Hölder theorem, the theorem stating that every linearly ordered group that satisfies an Archimedean property is isomorphic to a subgroup of the additive group (mathematics), group of real numbers, the classification of simple groups of order up to 200 (number), 200, the Automorphisms_of_the_symmetric_and_alternating_groups#The_exceptional_outer_automorphism_of_S6, anomalous outer automorphisms of the symmetric group S6, and Hölder's theorem, which implies that the Gamma function satisfies no algebraic differential equation. Another idea related to his name is the Hölder condition (or Hö ... [...More Info...]       [...Related Items...] Camille Jordan Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at the École polytechnique. He was an engineer by profession; later in life he taught at the École polytechnique and the Collège de France, where he had a reputation for eccentric choices of notation. He is remembered now by name in a number of results: * The Jordan curve theorem, a topological result required in complex analysis * The Jordan normal form and the Jordan matrix in linear algebra * In mathematical analysis, Jordan measure (or ''Jordan content'') is an area measure that predates measure theory * In group theory, the Composition series, Jordan–Hölder theorem on composition series is a basic result. * Jordan's theorem on finite linear groups Jordan's work did much to bring Galois theory into the mainstream. He also invest ... [...More Info...]       [...Related Items...]