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52 relations: Abel–Ruffini theorem, Abelian group, Algebraic equation, Alternating group, Commutator subgroup, Composition series, Cyclic group, Direct product of groups, Feit–Thompson theorem, Finite group, Finitely generated group, Galois group, Galois theory, Group extension, Group homomorphism, Group isomorphism, Group theory, Homomorphism, If and only if, Integer, Matematicheskii Sbornik, Mathematical proof, Mathematics, NC (complexity), Nilpotent group, Normal subgroup, Nth root, Order (group theory), P-group, Perfect core, Polycyclic group, Polynomial, Prime number, Prosolvable group, Quintic function, Quotient group, Semidirect product, Sign (mathematics), Simple group, Solvable group, Subalgebra, Subgroup series, Supersolvable group, Surjective function, Sylow theorems, Symmetric group, Trivial group, Uncountable set, Variety (universal algebra), Virtually, ..., Wreath product, Z-group. Expand index (2 more) »
Abel–Ruffini theorem
In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no algebraic solution—that is, solution in radicals—to the general polynomial equations of degree five or higher with arbitrary coefficients.
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Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Algebraic equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.
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Alternating group
In mathematics, an alternating group is the group of even permutations of a finite set.
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Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
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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.
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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Direct product of groups
In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.
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Feit–Thompson theorem
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable.
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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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Finitely generated group
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.
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Galois group
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.
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Galois theory
In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.
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Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group.
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Group homomorphism
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
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Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
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Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
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Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
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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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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Matematicheskii Sbornik
Matematicheskii Sbornik (Математический сборник, abbreviated Mat. Sb.) is a peer reviewed Russian mathematical journal founded by the Moscow Mathematical Society in 1866.
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Mathematical proof
In mathematics, a proof is an inferential argument for a mathematical statement.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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NC (complexity)
In complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors.
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Nilpotent group
A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with.
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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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Nth root
In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x: where n is the degree of the root.
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Order (group theory)
In group theory, a branch of mathematics, the term order is used in two unrelated senses.
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P-group
In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order.
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Perfect core
In mathematics, in the field of group theory, the perfect core (or perfect radical) of a group is its largest perfect subgroup.
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Polycyclic group
In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated).
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Prosolvable group
In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups.
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Quintic function
In algebra, a quintic function is a function of the form where,,,, and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero.
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
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Semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.
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Sign (mathematics)
In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.
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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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Solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions.
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Subalgebra
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
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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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Supersolvable group
In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups.
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Surjective function
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
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Sylow theorems
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains.
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Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
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Trivial group
In mathematics, a trivial group is a group consisting of a single element.
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Uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.
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Variety (universal algebra)
In the mathematical subject of universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities.
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Virtually
In mathematics, especially in the area of abstract algebra which studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index.
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Wreath product
In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product.
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Z-group
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups.
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Redirects here:
Derived length, Insoluble group, Soluble Group, Soluble group, Solvable groups, Solvable subgroup, Supersovable group.
References
[1] https://en.wikipedia.org/wiki/Solvable_group
