Lectures related to Group Theory: Combinatorial Approach

Topology: Free Groups and Their Properties

Discusses the theory of free groups, their properties, and relationships with other algebraic structures.

Coxeter Groups: Generators and Relations

Explores Coxeter groups, emphasizing generators, relations, and unique presentations in group theory.

Combinatorial Group Theory

Introduces combinatorial group theory, emphasizing group presentations, normal subgroups, and group quotients.

Bar Construction: Homology Groups and Classifying Space

Covers the bar construction method, homology groups, classifying space, and the Hopf formula.

Finite Abelian Groups: Classification Theorem

Presents the classification of finite abelian groups as products of cyclic groups, a fundamental result in various branches of mathematics.

Existence of Sylow Subgroups: Proof by Recurrence

Demonstrates the existence of p-Sylow subgroups in any finite group using induction and the abelian case.

Resoluble Groups: Quotients of Group

Covers the definition of solvable groups and the abelian property of group quotients.

Universal Property of Groups

Explores the universal property of groups, defining homomorphisms and exploring subgroup properties.

Abelian p-groups: Groupes abéliens

Focuses on abelian p-groups, demonstrating technical results for classifying finite abelian groups.

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Endomorphisms and automorphisms of totally disconnected locally compact groups

Explores endomorphisms and automorphisms of totally disconnected locally compact groups, emphasizing the properties of flat groups and free abelian subgroups.

Free Abelian Groups: Group Theory

Explores the concept of free abelian groups as an important left adjoint functor.

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Group Presentations

Covers the concept of group presentations using generators and relations to define groups.

Finitely Generated Abelian Groups

Explores finitely generated abelian groups, focusing on their structure and isomorphism theorems.

Free Product: Group Theory

Covers the concept of the free product in group theory and how to determine monomorphisms and homomorphisms.

Symmetry and Group Theory

Explores chirality, group completeness, Abelian groups, conjugate classes, and isomorphic groups in symmetry and group theory.

Abelian Groups: Free Abelian Groups

Explores free abelian groups, homomorphisms, and exact sequences in group theory.

Subgroups and Cosets: Lagrange's Theorem

Explores subgroups, normal subgroups, cosets, and Lagrange's theorem in group theory, emphasizing the importance of left cosets.

Introduction, Ch V: Sylow Subgroups

Explores the study of finite groups through subgroup analysis, focusing on Sylow subgroups.