MU-204 Group Theory

Unit · year 2

MU-204 · Group Theory

Threads structure25 lectures6 theorems

The mathematics of symmetry, from Lagrange's theorem to the Sylow theorems.

PREREQUISITES

L01	Symmetry and the Group Axioms —
L02	Examples: Cyclic, Dihedral, and Symmetric Groups —
L03	Subgroups and the Subgroup Test —
L04	Cyclic Groups and Their Subgroups —
L05	Cosets and Partitions Lagrange's theorem
L06	Lagrange's Theorem Lagrange's theorem
L07	Consequences: Element Orders and Groups of Prime Order Lagrange's theorem
L08	Normal Subgroups —
L09	Quotient Groups —
L10	Homomorphisms and Kernels The first isomorphism theorem
L11	The First Isomorphism Theorem The first isomorphism theorem
L12	The Second and Third Isomorphism Theorems The first isomorphism theorem
L13	Direct Products —
L14	Group Actions The orbit–stabiliser theorem
L15	Orbits and Stabilisers The orbit–stabiliser theorem
L16	The Orbit–Stabiliser Theorem The orbit–stabiliser theorem
L17	Conjugacy Classes and the Class Equation The orbit–stabiliser theorem
L18	Cauchy's Theorem Cauchy's theorem
L19	p-Groups and Their Centres Cauchy's theorem
L20	The Sylow Theorems The Sylow theorems
L21	Applying Sylow: Classifying Small Groups The Sylow theorems
L22	The Structure of Finite Abelian Groups Structure of finite abelian groups
L23	The Symmetric and Alternating Groups —
L24	Simplicity of Aₙ and Composition Series —
L25	Synthesis: Structure Forced by Counting Lagrange's theorem, The Sylow theorems