**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Inner automorphism

Summary

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. They can be realized via simple operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group.
If G is a group and g is an element of G (alternatively, if G is a ring, and g is a unit), then the function
is called (right) conjugation by g (see also conjugacy class). This function is an endomorphism of G: for all
where the second equality is given by the insertion of the identity between and Furthermore, it has a left and right inverse, namely Thus, is bijective, and so an isomorphism of G with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.
When discussing right conjugation, the expression is often denoted exponentially by This notation is used because composition of conjugations satisfies the identity: for all This shows that right conjugation gives a right action of G on itself.
The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of G is a group, the inner automorphism group of G denoted Inn(G).
Inn(G) is a normal subgroup of the full automorphism group Aut(G) of G. The outer automorphism group, Out(G) is the quotient group
The outer automorphism group measures, in a sense, how many automorphisms of G are not inner. Every non-inner automorphism yields a non-trivial element of Out(G), but different non-inner automorphisms may yield the same element of Out(G).
Saying that conjugation of x by a leaves x unchanged is equivalent to saying that a and x commute:
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 (or ring).

Official source

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 publications (18)

Related people (4)

Related lectures (50)

Related concepts (16)

Related courses (4)

Ontological neighbourhood

Groups and Products

Covers the concepts of groups, products, and homomorphisms, focusing on the hx set and group alternates.

Morphisms of Coverings

Covers the concept of morphisms of coverings and emphasizes the recognition of automorphisms.

Isomorphism of Groups

Covers bijections between automorphisms and equivariant bijections in group isomorphisms.

For a group G generated by k elements, the Nielsen equivalence classes are defined as orbits of the action of AutF(k), the automorphism group of the free group of rank k, on the set of generating k-tuples of G. Let p >= 3 be prime and G(p) the Gupta-Sidki ...

We discuss anomalies associated with outer automorphisms in gauge theories based on classical groups, namely charge conjugations for SU(N) and parities for SO(2r). We emphasize the inequivalence (yet related by a flavor transformation) between two versions ...

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 ...

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

Conjugacy class

In mathematics, especially group theory, two elements and of a group are conjugate if there is an element in the group such that This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under for all elements in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure.

Center (group theory)

In abstract algebra, the center of a group, G, is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning center. In set-builder notation, Z(G) = . The center is a normal subgroup, Z(G) ⊲ G. As a subgroup, it is always characteristic, but is not necessarily fully characteristic. The quotient group, G / Z(G), is isomorphic to the inner automorphism group, Inn(G). A group G is abelian if and only if Z(G) = G. At the other extreme, a group is said to be centerless if Z(G) is trivial; i.

PHYS-314: Quantum physics II

The aim of this course is to familiarize the student with the concepts, methods and consequences of quantum physics.

MATH-506: Topology IV.b - cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

MATH-110(a): Advanced linear algebra I

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux de ce sujet.