Inner automorphism - Citizendia

In abstract algebra, an inner automorphism of a group G is a function

f : G → G

defined by

f(x) = axa⁻¹, for all x in G,

where a is a given fixed element of G. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules 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 The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

The operation axa⁻¹ is called conjugation (see also conjugacy class). In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class Informally, in a conjugation a certain operation is applied, then another one (x) is carried out, and then the initial operation is reversed. Sometimes this matters ('take off shoes, take off socks, replace shoes'), and sometimes ('take off left glove, take off right glove, replace left glove') it doesn't.

In fact

axa⁻¹ = x

is equivalent to saying

ax = xa.

Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that In Mathematics, commutativity is the ability to change the order of something without changing the end result This is one good reason to study this concept in group theory.

1 Notation
2 Properties
3 The inner and outer automorphism groups
- 3.1 Types of groups
4 Ring case
5 Lie algebra case
6 Extension

Notation

The expression axa⁻¹ is often denoted exponentially by ^ax. This notation is used because we have the rule ^a(^bx)=^abx (giving a left action of G on itself). In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. An alternative form, leading to a right action, can be obtained by denoting a⁻¹xa as x^a.

Properties

Every inner automorphism is indeed an automorphism of the group G, i. 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 e. it is a bijective map from G to G and it is a homomorphism (meaning ^a(xy) = ^ax^ay). In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function

The inner and outer automorphism groups

The composition of two inner automorphisms is again an inner automorphism (as mentioned above: ^a(^bx)=^abx), and with this operation, the collection of all inner automorphisms of G is itself a group, the inner automorphism group of G denoted Inn(G). In Mathematics, a composite function represents the application of one function to the results of another

An automorphism of G which is not inner is called an outer automorphism.

Inn(G) is a normal subgroup of the full automorphism group Aut(G) of G. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself The quotient group

Aut(G)/Inn(G)

is known as the outer automorphism group Out(G). In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner The outer automorphism group measures, in a sense, how many automorphisms of G are not inner. Note however that the elements of Out(G) are not the outer automorphisms (which do not form a group) but are cosets of automorphisms. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH Every outer automorphism yields a non-trivial element of Out(G), but different outer automorphisms may yield the same element of Out(G).

By associating the element a in G with the inner automorphism f(x) = ^ax in Inn(G) as above, one obtains an isomorphism between the quotient group G/Z(G) (where Z(G) is the center of G) and the inner automorphism group:

G/Z(G) = Inn(G). 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 In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the

This is a consequence of the first isomorphism theorem, because Z(G) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism (conjugation changes nothing). In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural

Types of groups

It follows that the group Inn(G) of inner automorphisms is itself trivial (i. e. consists only of the identity element) if and only if G is abelian. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that ↔ An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

Inn(G) can only be a cyclic group when it is trivial, by a basic result on the center of a group. 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

At the opposite end of the spectrum, it is possible that the inner automorphisms exhaust the entire automorphism group; a group whose automorphisms are all inner is called 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

If the inner automorphism group of a perfect group G is simple, then G is called quasisimple. In Mathematics, in the realm of Group theory, a group is said to be perfect if it equals its own Commutator subgroup, or equivalently if the In Mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension E of

Ring case

Given a ring R and a unit u in R, the map f(x) = uxu^-1 is a ring automorphism of R. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

Lie algebra case

An automorphism of a Lie algebra $\mathfrak{g}$ is called an inner automorphism if it is of the form Ad_g, where Ad is the adjoint map and g is an element of a Lie group whose Lie algebra is $\mathfrak{g}$ . In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.

Extension

If G arises as the group of units of a ring A, then an inner automorphism on G can be extended to a projectivity on the projective space over A by inversive ring geometry. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, inversive ring geometry is the extension to the context of Associative rings of the concepts of Projective line, Homogeneous In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which In Mathematics, inversive ring geometry is the extension to the context of Associative rings of the concepts of Projective line, Homogeneous In particular, the inner automorphisms of the classical linear groups can be so extended. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation

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