Sporadic groups | Theorems in algebra | Finite groups | Mathematical classification systems | Group theory
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution. Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof. (Wikipedia).
GT23. Composition and Classification
Abstract Algebra: We use composition series as another technique for studying finite groups, which leads to the notion of solvable groups and puts the focus on simple groups. From there, we survey the classification of finite simple groups and the Monster group.
From playlist Abstract Algebra
Simple Groups - Abstract Algebra
Simple groups are the building blocks of finite groups. After decades of hard work, mathematicians have finally classified all finite simple groups. Today we talk about why simple groups are so important, and then cover the four main classes of simple groups: cyclic groups of prime order
From playlist Abstract Algebra
Simple groups, Lie groups, and the search for symmetry II | Math History | NJ Wildberger
This is the second video in this lecture on simple groups, Lie groups and manifestations of symmetry. During the 19th century, the role of groups shifted from its origin in number theory and the theory of equations to its role in describing symmetry in geometry. In this video we talk abou
From playlist MathHistory: A course in the History of Mathematics
Simple groups, Lie groups, and the search for symmetry I | Math History | NJ Wildberger
During the 19th century, group theory shifted from its origins in number theory and the theory of equations to describing symmetry in geometry. In this video we talk about the history of the search for simple groups, the role of symmetry in tesselations, both Euclidean, spherical and hyper
From playlist MathHistory: A course in the History of Mathematics
Every Group is a Quotient of a Free Group
First isomorphism theorem: https://youtu.be/ssVIJO5uNeg An explanation of a proof that every finite group is a quotient of a free group. A similar proof also applies to infinite groups because we can consider a free group on an infinite number of elements! Group Theory playlist: https://
From playlist Group Theory
Visual Group Theory, Lecture 5.7: Finite simple groups
Visual Group Theory, Lecture 5.7: Finite simple groups A group is said to be simple if its only normal subgroups are itself and the identity. Using Sylow theorems, we can frequently conclude statemens such as "there are no simple groups of order k", for some fixed k. After we provide seve
From playlist Visual Group Theory
Pierre Py - Complex geometry and higher finiteness properties of groups
Following C.T.C. Wall, we say that a group G is of type if it has a classifying space (a K(G,1)) whose n-skeleton is finite. When n=1 (resp. n=2) one recovers the condition of finite generation (resp. finite presentation). The study of examples of groups which are of type Fn-1 but not of
From playlist Geometry in non-positive curvature and Kähler groups
Cyclic groups and finite groups
Jacob goes into detail on some particularly important finite groups, and explains how groups and subgroups can be generated by their elements, along with some important consequences.
From playlist Basics: Group Theory
Abstract Algebra - 3.1 Finite Groups and Subgroups: Terminology and Notation
Most of this chapter will revolve around the idea of a subgroup. However, we must begin by being able to differentiate between a finite group and infinite group. We look at some notation and definitions (order of a group, order of an element) before jumping into subgroups. Video Chapters:
From playlist Abstract Algebra - Entire Course
Julia Plavnik: "Classifying small fusion categories"
Actions of Tensor Categories on C*-algebras 2021 "Classifying small fusion categories" Julia Plavnik - Indiana University, Mathematics Abstract: Classifying fusion categories is a problem that at the moment seems out of reach, since it includes the classification of finite groups and sem
From playlist Actions of Tensor Categories on C*-algebras 2021
Kristin Courtney: "The abstract approach to classifying C*-algebras"
Actions of Tensor Categories on C*-algebras 2021 Mini Course: "The abstract approach to classifying C*-algebras" Kristin Courtney - Westfälische Wilhelms-Universität Münster Institute for Pure and Applied Mathematics, UCLA January 21, 2021 For more information: https://www.ipam.ucla.edu
From playlist Actions of Tensor Categories on C*-algebras 2021
8ECM Invited Lecture: Stuart White
From playlist 8ECM Invited Lectures
On the classification of fusion categories – Sonia Natale – ICM2018
Algebra Invited Lecture 2.5 On the classification of fusion categories Sonia Natale Abstract: We report, from an algebraic point of view, on some methods and results on the classification problem of fusion categories over an algebraically closed field of characteristic zero. © Interna
From playlist Algebra
This is an informal talk on sporadic groups given to the Archimedeans (the Cambridge undergraduate mathematical society). It discusses the classification of finite simple groups and some of the sporadic groups, and finishes by briefly describing monstrous moonshine. For other Archimedeans
From playlist Math talks
Nuclear C*-algebras: From quasidiagonality to classification and back again – W. Winter – ICM2018
Analysis and Operator Algebras Invited Lecture 8.20 Structure of nuclear C*-algebras: From quasidiagonality to classification and back again Wilhelm Winter Abstract: I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-alge
From playlist Analysis & Operator Algebras
Christopher Schafhauser: On the classification of nuclear simple C*-algebras, Lecture 3
Mini course of the conference YMC*A, August 2021, University of Münster. Abstract: A conjecture of George Elliott dating back to the early 1990’s asks if separable, simple, nuclear C*-algebras are determined up to isomorphism by their K-theoretic and tracial data. Restricting to purely i
From playlist YMC*A 2021
Regular permutation groups and Cayley graphs
Cheryl Praeger (University of Western Australia). Plenary Lecture from the 1st PRIMA Congress, 2009. Plenary Lecture 11. Abstract: Regular permutation groups are the 'smallest' transitive groups of permutations, and have been studied for more than a century. They occur, in particular, as
From playlist PRIMA2009
This lecture is part of an online math course on group theory. We review free abelian groups, then construct free (non-abelian) groups, and show that they are given by the set of reduced words, and as a bonus find that they are residually finite.
From playlist Group theory