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Composition series

Index Composition series

In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. [1]

37 relations: Abelian category, Abstract algebra, Algebraic structure, Artinian module, Artinian ring, Bulletin of the American Mathematical Society, Camille Jordan, Category (mathematics), Chief series, Direct sum of modules, Filtered algebra, Finite group, Fundamental theorem of arithmetic, Glossary of category theory, Group (mathematics), Group with operators, If and only if, Infinite group, Inner automorphism, Isomorphism, Isomorphism class, Krohn–Rhodes theory, Maximal and minimal elements, Module (mathematics), Noetherian module, Normal subgroup, Otto Hölder, Permutation, Schreier refinement theorem, Semisimple module, Simple group, Simple module, Subgroup series, Subobject, Transfinite induction, Up to, Zassenhaus lemma.

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

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Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

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Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.

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Artinian module

In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules.

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Artinian ring

In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals.

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Bulletin of the American Mathematical Society

The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.

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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.

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Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

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Chief series

In abstract algebra, a chief series is a maximal normal series for a group.

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Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.

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Filtered algebra

In mathematics, a filtered algebra is a generalization of the notion of a graded algebra.

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Finite group

In abstract algebra, a finite group is a mathematical group with a finite number of elements.

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Fundamental theorem of arithmetic

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

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Glossary of category theory

This is a glossary of properties and concepts in category theory in mathematics.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Group with operators

In abstract algebra, a branch of pure mathematics, the algebraic structure group with operators or Ω-group can be viewed as a group with a set Ω that operates on the elements of the group in a special way.

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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Infinite group

In group theory, an area of mathematics, an infinite group is a group, of which the underlying set contains an infinite number of elements.

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Inner automorphism

In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element.

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Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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Isomorphism class

An isomorphism class is a collection of mathematical objects isomorphic to each other.

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Krohn–Rhodes theory

In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components.

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Maximal and minimal elements

In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.

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Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.

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Noetherian module

In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.

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Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.

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Otto Hölder

Otto Ludwig Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.

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Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

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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 isomorphic one.

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Semisimple module

In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts.

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Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

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Simple module

In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that have no non-zero proper submodules.

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Subgroup series

In mathematics, specifically group theory, a subgroup series is a chain of subgroups: Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups.

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Subobject

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category.

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Transfinite induction

Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.

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Up to

In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.

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Zassenhaus lemma

In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice.

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Redirects here:

Composition factor, Composition length, Jordan Holder theorem, Jordan holder theorem, Jordan-Hoelder Theorem, Jordan-Hoelder decomposition, Jordan-Hoelder theorem, Jordan-Holder Theorem, Jordan-Holder decomposition, Jordan-Holder theorem, Jordan-Hölder Theorem, Jordan-Hölder decomposition, Jordan-Hölder theorem, Jordan–Hölder decomposition, Jordan–Hölder theorem.

References

[1] https://en.wikipedia.org/wiki/Composition_series

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