# Automorphism group

In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group is the group consisting of all group automorphisms of X. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. (Wikipedia).

Group automorphisms in abstract algebra

Group automorphisms are bijective mappings of a group onto itself. In this tutorial I define group automorphisms and introduce the fact that a set of such automorphisms can exist. This set is proven to be a subgroup of the symmetric group. You can learn more about Mathematica on my Udem

From playlist Abstract algebra

Visual Group Theory, Lecture 6.2: Field automorphisms

Visual Group Theory, Lecture 6.2: Field automorphisms A field automorphism is a structure preserving map from a field F to itself. This means that it must be both a homomorphism of both the addtive group (F,+) and the multiplicative group (F-{0},*). We show that any automorphism of an ext

From playlist Visual Group Theory

Automorphism groups and modular arithmetic

Jacob explains the concept of the automorphism group of a group, as well as how such groups give rise to useful properties of multiplication in modular arithmetic, including Fermat's Little Theorem.

From playlist Basics: Group Theory

Homomorphisms in abstract algebra

In this video we add some more definition to our toolbox before we go any further in our study into group theory and abstract algebra. The definition at hand is the homomorphism. A homomorphism is a function that maps the elements for one group to another whilst maintaining their structu

From playlist Abstract algebra

Group automorphism example

In this tutorial I present the cyclic group of three elements as a group automorphism. You can learn more about Mathematica on my Udemy courses: https://www.udemy.com/mathematica/ https://www.udemy.com/mathematica-for-statistics/

From playlist Abstract algebra

Visual Group Theory, Lecture 4.6: Automorphisms

Visual Group Theory, Lecture 4.6: Automorphisms An automorphism is an isomorphism from a group to itself. The set of all automorphisms of G forms a group under composition, denoted Aut(G). After a few simple examples, we learn how Aut(Z_n) is isomorphic to U(n), which is the group consist

From playlist Visual Group Theory

Abstract Algebra - 6.5 Automorphisms

We finish up chapter 6 by discussion automorphisms and inner automorphisms. An automorphism is just a special isomorphism that maps a group to itself. An inner-automorphism uses conjugation of an element and its inverse to create a mapping. Video Chapters: Intro 0:00 What is an Automorphi

From playlist Abstract Algebra - Entire Course

GT11. Group Automorphisms

EDIT: At 3:20, nonzero elements have order 3, not 2. Abstract Algebra: We consider the group Aut(G) of automorphisms of G, the isomorphisms from G to itself. We show that the inner automorphisms of G, induced by conjugation, form a normal subgroup Inn(G) of Aut(G), and that Inn(G) is i

From playlist Abstract Algebra

Example of Group Automorphism 1 (Requires Linear Algebra)

Matrix Theory: We compute the automorphism groups of G = Z/10 and G=Z/2 x Z/2. The first case is a warm up for Part 2. The second case can be recast as a linear algebra problem with matrix groups.

From playlist Matrix Theory

Jason Parker - Covariant Isotropy of Grothendieck Toposes

Talk at the school and conference “Toposes online” (24-30 June 2021): https://aroundtoposes.com/toposesonline/ Slides: https://aroundtoposes.com/wp-content/uploads/2021/07/ParkerSlidesToposesOnline.pdf Covariant isotropy can be regarded as providing an abstract notion of conjugation or i

From playlist Toposes online

Zlil Sela - Automorphisms of groups and a higher rank JSJ decomposition

The JSJ (for groups) was originally constructed to study the automorphisms and the cyclic splittings of a (torsion-free) hyperbolic group. Such a structure theory was needed to complete the solution of the isomorphism problem for (torsion-free) hyperbolic groups. Later, the JSJ was genera

From playlist Geometry in non-positive curvature and Kähler groups

Alexander HULPKE - Computational group theory, cohomology of groups and topological methods 5

The lecture series will give an introduction to the computer algebra system GAP, focussing on calculations involving cohomology. We will describe the mathematics underlying the algorithms, and how to use them within GAP. Alexander Hulpke's lectures will being with some general computation

Group theory 30: Outer automorphisms

This lecture is part of an online course on group theory. We find the automorphism groups of symmetric groups, and in particular show that the symmetric group on 6 points has "extra" (outer) automorphisms.

From playlist Group theory

Non-amenable groups admitting no sofic approximation by expander graphs - Gabor Kun

Stability and Testability Topic: Non-amenable groups admitting no sofic approximation by expander graphs Speaker: Gabor Kun Affiliation: Alfréd Rényi Institute of Mathematics Date: February 10, 2021 For more video please visit http://video.ias.edu

From playlist Stability and Testability

Karen Vogtmann: The geometry and topology of automorphism groups of free groups

HYBRID EVENT Recorded during the meeting "Groups Acting on Fractals" the April 11, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Ma

From playlist Topology

10/13/17 Yuri Berest

Differential Isomorphism and Equivalence of Algebraic Varieties Board at 49:35 Sum_i=1^N 2/(x-phi_i(y,t))^2

From playlist Fall 2017

Group Isomorphisms in Abstract Algebra

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Group Isomorphisms in Abstract Algebra - Definition of a group isomorphism and isomorphic groups - Example of proving a function is an Isomorphism, showing the group of real numbers under addition is isomorphic to the group of posit

From playlist Abstract Algebra

CTNT 2022 - An Introduction to Galois Representations (Lecture 1) - by Alvaro Lozano-Robledo

This video is part of a mini-course on "An Introduction to Galois Representations" that was taught during CTNT 2022, the Connecticut Summer School and Conference in Number Theory. More about CTNT: https://ctnt-summer.math.uconn.edu/