Concept

Automorphisms of the symmetric and alternating groups

In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements. , and thus . Formally, is complete and the natural map is an isomorphism. , and the outer automorphism is conjugation by an odd permutation. Indeed, the natural maps are isomorphisms. trivial: , and is a semidirect product. , and Among symmetric groups, only S6 has a non-trivial outer automorphism, which one can call exceptional (in analogy with exceptional Lie algebras) or exotic. In fact, Out(S6) = C2. This was discovered by Otto Hölder in 1895. The specific nature of the outer automorphism is as follows: the sole identity permutation maps to itself; a 2-cycle such as (1 2) maps to the product of three 2-cycles such as (1 2)(3 4)(5 6) and vice versa, there being 15 permutations each way; a 3-cycle such as (1 2 3) maps to the product of two 3-cycles such as (1 4 5)(2 6 3) and vice versa, accounting for 40 permutations each way; a 4-cycle such as (1 2 3 4) maps to another 4-cycle such as (1 6 2 4) accounting for 90 permutations; the product of two 2-cycles such as (1 2)(3 4) maps to another product of two 2-cycles such as (3 5)(4 6), accounting for 45 permutations; a 5-cycle such as (1 2 3 4 5) maps to other 5-cycles such as (1 3 6 5 2) accounting for 144 permutations; the product of a 2-cycle and a 3-cycle such as (1 2 3)(4 5) maps to a 6-cycle such as (1 2 5 3 4 6) and vice versa, accounting for 120 permutations each way; the product of a 2-cycle and a 4-cycle such as (1 2 3 4)(5 6) maps to another such permutation such as (1 4 2 6)(3 5) accounting for the 90 remaining permutations. Thus, all 720 permutations on 6 elements are accounted for. The outer automorphism does not preserve cycle structure in general, mapping single cycles to the product of two cycles and vice versa.

Official source

https://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (1)

MATH-113: Algebra I - fundamental structures

Le but de ce cours est d'introduire et d'étudier les notions de base de l'algèbre abstraite.

Related concepts (4)

Almost simple group

In mathematics, a group is said to be almost simple if it contains a non-abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group A is almost simple if there is a (non-abelian) simple group S such that Trivially, non-abelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group.

Outer automorphism group

In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete. An automorphism of a group that is not inner is called an outer automorphism. The cosets of Inn(G) with respect to outer automorphisms are then the elements of Out(G); this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups.

Mathieu group

In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M11, M12, M22, M23 and M24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic groups to be discovered. Sometimes the notation M9, M10, M20 and M21 is used for related groups (which act on sets of 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups.

Official source

https://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups

About this result

Related courses (1)

MATH-113: Algebra I - fundamental structures

Le but de ce cours est d'introduire et d'étudier les notions de base de l'algèbre abstraite.

Related lectures (27)

Automorphism Groups: Trees and Graphs III

Explores automorphism groups of trees and graphs, including actions on trees and group homomorphisms.

Automorphism groups of trees and graphs

Explores automorphisms of graphs, focusing on automorphism groups, Cayley-Abels graphs, and quasi-isometry.

Automorphism groups: Trees and Graphs

Explores automorphism groups in trees and graphs, focusing on ends and types of automorphisms.

Related publications (9)

Symmetry in design and decoding of polar-like codes

Kirill Ivanov

The beginning of 21st century provided us with many answers about how to reach the channel capacity. Polarization and spatial coupling are two techniques for achieving the capacity of binary memoryless symmetric channels under low-complexity decoding algor ...

EPFL2022

On the Efficiency of Polar-Like Decoding for Symmetric Codes

Rüdiger Urbanke, Kirill Ivanov

The recently introduced polar codes constitute a breakthrough in coding theory due to their capacity-achieving property. This goes hand in hand with a quasilinear construction, encoding, and successive cancellation list decoding procedures based on the Plo ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

Arithmetically rigid schemes via deformation theory of equivariant vector bundles

Maciej Emilian Zdanowicz

We analyze the deformation theory of equivariant vector bundles. In particular, we provide an effective criterion for verifying whether all infinitesimal deformations preserve the equivariant structure. As an application, using rigidity of the Frobenius ho ...

SPRINGER HEIDELBERG2021

Related concepts (4)

Almost simple group

Outer automorphism group

Mathieu group