Group automorphisms | Group theory
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 which 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. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group. For example, for the alternating group, An, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering An as a subgroup of the symmetric group, Sn, conjugation by any odd permutation is an outer automorphism of An or more precisely "represents the class of the (non-trivial) outer automorphism of An", but the outer automorphism does not correspond to conjugation by any particular odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element. (Wikipedia).
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
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
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
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
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
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
Kernel of a group homomorphism
In this video I introduce the definition of a kernel of a group homomorphism. It is simply the set of all elements in a group that map to the identity element in a second group under the homomorphism. The video also contain the proofs to show that the kernel is a normal subgroup.
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
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
Karen Vogtmann - Outer space for right-angled Artin groups
Karen Vogtman (Cornell University, USA) Right-angled Artin groups interpolate between free groups and free abelian groups, so one may think of their outer automorphism groups as interpolating between Out(F_n) and GL(n,Z). I will describe an Outer space for these automorphism groups which
From playlist T1-2014 : Random walks and asymptopic geometry of groups.
Stefaan Vaes: "Outer actions of amenable groups on von Neumann algebras"
Actions of Tensor Categories on C*-algebras 2021 Mini Course: "Outer actions of amenable groups on von Neumann algebras" Stefaan Vaes - KU Leuven Abstract: I will give a survey lecture on the classification of outer actions of amenable groups on von Neumann algebras with the main focus b
From playlist Actions of Tensor Categories on C*-algebras 2021
Karen VOGTMANN - Spaces of Graphs, Tori and Other Flat Gamma-complexes
Spaces of finite graphs play a key role in perturbative quantum field theory, but also in many other areas of science and mathematics. Among these is geometric group theory, where they are used to model groups of automorphism of free groups. Graphs can be thought of as 1-dimensional flat m
From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday
Camille Horbez: Growth under random products of automorphisms of a free group
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Algebra
These are abstract constructions needed to define nonabelian H2
From playlist Stacks
Vincent Guirardel: Natural subgroups of automorphisms
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Algebra
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
Michael BORINSKY - The Euler Characteristic of Out(Fn) and the Hopf Algebra of Graphs
In their 1986 work, Harer and Zagier gave an expression for the Euler characteristic of the moduli space of curves, M_gn, or equivalently the mapping class group of a surface. Recently, in joint work with Karen Vogtmann, we performed a similar analysis for Out(Fn), the outer automorphism g
From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday
Abstract Algebra | The inner automorphisms of a group.
http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Abstract Algebra