Outer automorphism group

In mathematics, the outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G 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. In Mathematics, a group G is said to be complete if every Automorphism of G is inner, and the group is a centerless group

Note that the elements of Out(G) are cosets of automorphisms of G, and not themselves automorphisms. This is an instance of the fact that quotients of groups are not in general subgroups. In practice, however, elements of Aut(G) which are not inner automorphisms are often called outer automorphisms; they are representatives of the non-trivial cosets in Out(G).

The Schreier conjecture asserts that Out(G) is always a solvable group when G is a finite simple group. In finite group theory, the Schreier conjecture asserts that the group of Outer automorphisms of every finite simple group is solvable. In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning This result is now known to be true as a corollary of the classification of finite simple groups, although no simpler proof is known. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups.

This group is important in the topology of surfaces because there is a happy connection provided by the Dehn-Nielsen theorem: the extended mapping class group of the surface is the Out of its fundamental group. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Mathematics, in the sub-field of Geometric topology, the mapping class group is an important algebraic invariant of a Topological space. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

1 Out(G) for some finite groups
2 The outer automorphisms of the symmetric and alternating groups
3 Outer automorphism groups of complex Lie groups
4 Puns
5 External links
6 See also

Out(G) for some finite groups

For the outer automorphism groups of all finite simple groups see the list of finite simple groups. In Mathematics, the Classification of finite simple groups states thatevery finite Simple group is cyclic, or alternating, or in one of 16 families Sporadic simple groups and alternating groups (other than the alternating group A₆; see below) all have outer automorphism groups of order 1 or 2. The outer automorphism group of a finite simple group of Lie type is an extension of a group of "diagonal automorphisms" (cyclic except for D_n(q) when it has order 4), a group of "field automorphisms" (always cyclic), and a group of "graph automorphisms" (of order 1 or 2 except for D₄(q) when it is the symmetric group on 3 points). In Mathematics, a group of Lie type G(k is a (not necessarily finite group of rational points of a reductive Linear algebraic group G with In Mathematics, the Classification of finite simple groups states thatevery finite Simple group is cyclic, or alternating, or in one of 16 families These extensions are semidirect products except that for the Suzuki-Ree groups the graph automorphism squares to a generator of the field automorphisms. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, a group of Lie type G(k is a (not necessarily finite group of rational points of a reductive Linear algebraic group G with

Group	Parameter	Out(G)	$\| Out(G) \|$
Z	infinite cyclic	Z₂	2
Z_n	n > 2	Z_n^×	φ(n) = $n\prod_{p\|n}\left(1-\frac{1}{p}\right)$ elements
Z_pⁿ	p prime, n > 1	GL_n(p)	(pⁿ - 1)(pⁿ - p )(pⁿ - p²) . The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Modular arithmetic the set of Congruence classes Relatively prime to the modulus n form a group under multiplication called the multiplicative In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation . . (pⁿ - p^n-1) elements
S_n	n not equal to 6	trivial	1
S₆		Z₂ (see below)	2
A_n	n not equal to 6	Z₂	2
A₆		Z₂ × Z₂(see below)	4
PSL₂(p)	p > 3 prime	Z₂	2
PSL₂(2ⁿ)	n > 1	Z_n	n
PSL₃(4) = M₂₁		Dih₆	12
M_n	n = 11, 23, 24	trivial	1
M_n	n = 12, 22	Z₂	2
Co_n	n = 1, 2, 3	trivial	1

The outer automorphisms of the symmetric and alternating groups

For more details on this topic, see Automorphisms of the symmetric and alternating groups. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, a trivial group is a group consisting of a single element In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, an alternating group is the group of Even permutations of a Finite set. In Mathematics, an alternating group is the group of Even permutations of a Finite set. In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group In the Mathematical field of Group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections In the Mathematical field of Group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple In Mathematics, a trivial group is a group consisting of a single element In the Mathematical field of Group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple In Mathematics, the Conway groups Co1 Co2 and Co3 are three Sporadic groups discovered by John Horton Conway. In Mathematics, a trivial group is a group consisting of a single element In Group theory, a branch of Mathematics, the Automorphisms and Outer automorphisms of the Symmetric groups and Alternating groups are

The outer automorphism group of a finite simple group in some infinite family of finite simple groups can almost always be given by a uniform formula that works for all elements of the family. There is just one exception to this: the alternating group A₆ has outer automorphism group of order 4, rather than 2 for the other simple alternating groups (given by conjugation by an odd permutation). In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even Equivalently the symmetric group S₆ is the only symmetric group with a non-trivial outer automorphism group.

$\begin{align}n\neq 6: & {}\quad\mbox{Out}(S_n)=1 \\n\geq 3,\ n\neq 6: & {} \quad\mbox{Out}(A_n)=C_2 \\& {} \quad \mbox{Out}(S_6)=C_2 \\& {} \quad \mbox{Out}(A_6)=C_2 \times C_2\end{align}$

Outer automorphism groups of complex Lie groups

Let G now be a connected reductive group over an algebraically closed field. In Mathematics, a reductive group is an Algebraic group G such that the Unipotent radical of the Identity component of G In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients Then any two Borel subgroups are conjugate by an inner automorphism, so to study outer automorphisms it suffices to consider automorphisms that fix a given Borel subgroup. In the theory of Algebraic groups, a Borel subgroup of an Algebraic group G is a maximal Zariski closed and connected solvable Associated to the Borel subgroup is a set of simple roots, and the outer automorphism may permute them, while preserving the structure of the associated Dynkin diagram. This article discusses root systems in mathematics For root systems of Plants see Root. This article discusses root systems in mathematics For root systems of Plants see Root. In this way one may identify the automorphism group of the Dynkin diagram of G with a subgroup of Out(G).

$D 4$ has a very symmetric Dynkin diagram, which yields a large outer automorphism group of Spin(8), namely $Out(Spin(8)) = S 3$ ; this is called triality. In Mathematics, SO(8 is the Special orthogonal group acting on eight-dimensional Euclidean space. In Mathematics, triality is a relationship between three Vector spaces analogous to the duality relation between Dual vector spaces Most commonly

Puns

The term "Outer automorphism" lends itself to puns: the term outermorphism is sometimes used for "outer automorphism", and a particular geometry on which $\scriptstyle\operatorname{Out}(F_n)$ acts is called outer space. In Mathematics, specifically Geometric group theory, a geometric group action is a certain type of action of a Discrete group on a Metric In Mathematics, Out( Fn) is the Outer automorphism group of a Free group on n generators

External links

ATLAS of Finite Group Representations-V3

(contains a lot of information on various classes of finite groups (in particular sporadic simple groups), including the order of Out(G) for each group listed.

Contents

Out(G) for some finite groups

The outer automorphisms of the symmetric and alternating groups

Outer automorphism groups of complex Lie groups

Puns

External links

See also