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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. In particular, the finite symmetric group S n {\displaystyle \mathrm {S} _{n}} defined over a finite set of n {\displaystyle n} symbols consists of the…

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Overview

Definition and first properties

Applications

Group properties and special elements

Conjugacy classes

Low degree groups

Relation with alternating group

Generators and relations

Subgroup structure

Cyclic subgroups

Automorphism group

Homology

Representation theory

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Symmetric group

Nodes159
Edges158
Triples121
Avg. degree1.99
Density0.012579
Components1

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Symmetric group

Top relations

related to Normal subgroups · 20
Symmetric group → Any, Ch, Dixon, For, However, If, Klein, Mortimer, Onofri, Schreier, Schreier-Ulam, Scott, Since, Sn, That, The, This, Those, Ulam, Vitali
related to External links · 17
Symmetric group → Archived, EMS Press, Encyclopedia, Eric, June, Learning, Marcus, Mathematics, MathWorld, OEIS Entries, S4, Sautoy, Symmetric, Symmetry, Weisstein, Wikiversity, Wiktionary-logo-en-v2
has application · 13
Symmetric group → An, Bruhat, Coxeter, Galois, Higman, In, Lie, Schur, Sims, Subgroups, The, Weyl, Young
related to Maps between symmetric groups · 11
Symmetric group → A4, C1, Other, S0, S1, S2, S3, S4, S5, S6, Sn
related to Cyclic subgroups · 8
Symmetric group → Cyclic, For, Landau's, S5, Sn, The, This, When
related to Homology · 7
Symmetric group → Cp, One, S2, Sn, The, This, Thus
related to Representation theory · 7
Symmetric group → Its, Sn, The, Therefore, This, Unlike, Young
related to Sylow subgroups · 7
Symmetric group → AGL, Fp, Frobenius, The, The Sylow, There, They
related to Definition and first properties · 5
Symmetric group → For, If, Sigma, Sym, The
related to Multiplication · 5
Symmetric group → Applying, Concretely, For, So, The

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Important terminology

group symmetric sn displaystyle permutation set subgroup elements order groups subgroups finite permutations representation product theory transpositions automorphism every element

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Symmetric groupis aCoxeter group of type An and occurs as the Weyl group of the general linear group0.90text
Symmetric groupis aparticular case of the representation theory of finite groups0.90text
Galois theoryinstance ofwill mean a symmetric group on a finite set.The symmetric group is important to diverse areas of mathematics0.80text
invariant theoryinstance ofwill mean a symmetric group on a finite set.The symmetric group is important to diverse areas of mathematics0.80text
the representation theory of Lie groupsinstance ofwill mean a symmetric group on a finite set.The symmetric group is important to diverse areas of mathematics0.80text
and combinatoricsinstance ofwill mean a symmetric group on a finite set.The symmetric group is important to diverse areas of mathematics0.80text
Symmetric grouphas applicationThe0.60section
Symmetric grouphas applicationGalois0.60section
Symmetric grouphas applicationIn0.60section
Symmetric grouphas applicationLie0.60section
Symmetric grouphas applicationSchur0.60section
Symmetric grouphas applicationCoxeter0.60section

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