Sylow theorems

Visual Group Theory, Lecture 5.6: The Sylow theorems

Visual Group Theory, Lecture 5.6: The Sylow theorems The three Sylow theorems help us understand the structure of non-abelian groups by placing strong restrictions on their p-subgroups (i.e., subgroups of prime power order). The first Sylow theorem says that for every p^k dividing |G|=p^n

From playlist Visual Group Theory

GT20. Overview of Sylow Theory

Abstract Algebra: As an analogue of Cauchy's Theorem for subgroups, we state the three Sylow Theorems for finite groups. Examples include S3 and A4. We also note the analogue to Sylow Theory for p-groups. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-gr

From playlist Abstract Algebra

Group theory 14: Sylow theorems

This lecture is part of an online mathematics course on group theory. It gives the proofs of the Sylow theorems about the Sylow p-subgroups: those of order the largest power of p dividing the order of a group. Correction: Yenan Wang pointed out that at 18:18 D4 should be D8, the dihedral

From playlist Group theory

GT20.1. Sylow Theorems - Proofs

Abstract Algebra: We give proofs of the three Sylow Theorems. Techniques include the class equation and group actions on subgroups. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theory Master list at http://mathdoctorbob.org/UReddit.html

From playlist Abstract Algebra

Joe Neeman: Gaussian isoperimetry and related topics II

The Gaussian isoperimetric inequality gives a sharp lower bound on the Gaussian surface area of any set in terms of its Gaussian measure. Its dimension-independent nature makes it a powerful tool for proving concentration inequalities in high dimensions. We will explore several consequence

From playlist Winter School on the Interplay between High-Dimensional Geometry and Probability

Lie derivatives of differential forms

Introduces the lie derivative, and its action on differential forms. This is applied to symplectic geometry, with proof that the lie derivative of the symplectic form along a Hamiltonian vector field is zero. This is really an application of the wonderfully named "Cartan's magic formula"

From playlist Symplectic geometry and mechanics

Simple Group 168 - Sylow Theory - Part 1

Abstract Algebra: Let G be a simple group of order 168. We calculate the number of Sylow subgroups, number of elements of a given order, and conjugacy class structure. In Part 1, we consider Sylow-p subgroup for p = 3, 7.

From playlist Abstract Algebra

Joe Neeman: Gaussian isoperimetry and related topics III

The Gaussian isoperimetric inequality gives a sharp lower bound on the Gaussian surface area of any set in terms of its Gaussian measure. Its dimension-independent nature makes it a powerful tool for proving concentration inequalities in high dimensions. We will explore several consequence

From playlist Winter School on the Interplay between High-Dimensional Geometry and Probability

AKPotW: Normalizer of a p-Sylow Subgroup [Algebra]

A neat result about the normalizer of a p-sylow subgroup. For a written solution, check out the blog!

From playlist Center of Math: Problems of the Week

Sylow Theory for Order 12 Groups 1

Abstract Algebra: Let G be a finite group of order 12. We apply Sylow theory to study such groups. In Part 1, we consider the abelian cases and A4, the alternating group on 4 letters.

From playlist Abstract Algebra

Group theory 2: Cayley's theorem

This is lecture 2 of an online mathematics course on group theory. It describes Cayley's theorem that every abstract group is the group of symmetries of something, and as examples shows the Cayley graphs of the Klein 4-group and the symmetric group on 3 points.

From playlist Group theory

Simple Group 168 - Sylow Theory - Part 2

Note: Part 5 goes off the rails; I can't just assume the subgroup we choose normalizes H_2 a priori. We can still fix with elementary methods and the occasional lucky break. Fix for Part 5 (2:15) - disregard table: Key to note is that there are no elements of orders 6, 14, or 21 (s

From playlist Abstract Algebra

GT20.2 Sylow Theory for Simple 60

EDIT: At 6:50, 1, 3, 5, 7 should be 1, 3, 7, 9. At 9:35, n3 should be n2. Abstract Algebra: Using Sylow theory, we show that any simple, non-abelian group with 60 elements is isomorphic to A_5, the alternating group on 5 letters. As an application, we show that A_5 is isomorphic to t

From playlist Abstract Algebra

Joe Neeman: Gaussian isoperimetry and related topics I

The Gaussian isoperimetric inequality gives a sharp lower bound on the Gaussian surface area of any set in terms of its Gaussian measure. Its dimension-independent nature makes it a powerful tool for proving concentration inequalities in high dimensions. We will explore several consequence

From playlist Winter School on the Interplay between High-Dimensional Geometry and Probability

Sylow Theory for Order 12 Groups 2

Abstract Algebra: Let G be a finite group of order 12. Using Sylow Theory, we consider the isomorphism types of G when n_3 = 1 and n_1. In this case, G is isomorphic to either D_12, the symmetry group of a regular hexagon, or a nontrivial semidirect product of Z/3 and Z/4.

From playlist Abstract Algebra