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Alternating group

Index Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. [1]

42 relations: Abelian group, Automorphism, Commutator subgroup, Conjugacy class, Crelle's Journal, Cycle graph (algebra), Cyclic group, Dover Publications, Exceptional isomorphism, Factorial, Finite set, Galois theory, Group (mathematics), Group cohomology, Group homomorphism, Group of Lie type, Homology (mathematics), Icosahedral symmetry, If and only if, Index of a subgroup, Inner automorphism, Kernel (algebra), Klein four-group, Lagrange's theorem (group theory), Lodovico Ferrari, Mathematics, Normal subgroup, Outer automorphism group, Parity of a permutation, Perfect group, Permutation, Projective linear group, Quartic function, Resolvent (Galois theory), Schur multiplier, Simple group, Solvable group, Stable homotopy theory, Symmetric group, Symmetry group, Tetrahedral symmetry, Trivial group.

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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Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

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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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Conjugacy class

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.

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Crelle's Journal

Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik (in English: Journal for Pure and Applied Mathematics).

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Cycle graph (algebra)

In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.

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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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Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

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Exceptional isomorphism

In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such isomorphisms.

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Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

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Finite set

In mathematics, a finite set is a set that has a finite number of elements.

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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 (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 cohomology

In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology.

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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 of Lie type

In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field.

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Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

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Icosahedral symmetry

A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.

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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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Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).

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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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Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.

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Klein four-group

In mathematics, the Klein four-group (or just Klein group or Vierergruppe, four-group, often symbolized by the letter V or as K4) is the group, the direct product of two copies of the cyclic group of order 2.

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Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

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Lodovico Ferrari

Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian mathematician.

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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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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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Outer automorphism group

In mathematics, the outer automorphism group of a group,, is the quotient,, where is the automorphism group of and) is the subgroup consisting of inner automorphisms.

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Parity of a permutation

In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations.

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Perfect group

In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial).

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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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Projective linear group

In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V).

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Quartic function

In algebra, a quartic function is a function of the form where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial.

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Resolvent (Galois theory)

In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root.

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Schur multiplier

In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H2(G, Z) of a group G. It was introduced by in his work on projective representations.

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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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Stable homotopy theory

In mathematics, stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor.

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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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Symmetry group

In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.

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Tetrahedral symmetry

A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.

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Trivial group

In mathematics, a trivial group is a group consisting of a single element.

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Redirects here:

Alt(n), Alternating groups, Simplicity of the alternating groups.

References

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

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