Lectures related to Abelian p-groups: Groupes abéliens

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Abelian Groups: First Approach

Introduces the theory of abelian groups, focusing on p-abelian groups and their structure.

Abelian Groups: Technical Lemmas

Covers technical lemmas on abelian p-groups and maximal order elements.

Group Theory: Active Learning Session

Delves into group theory, emphasizing the centralizer of elements and the classification of finite abelian groups.

Sylow Subgroups: Structure and Properties

Explores the properties and structure of Sylow subgroups in group theory, emphasizing a theorem-independent approach.

Existence of Sylow Subgroups: Proof by Recurrence

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

Finite Abelian Groups & Rings

Explores the classification of finite abelian groups and rings, emphasizing elementary divisors and properties of rings.

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.

Introduction, Ch V: Sylow Subgroups

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

Algebra: Group Actions and Direct Products

Covers group actions, conjugacy classes, centralizers, direct products, and finite abelian groups.

Useful Lemmas for p-Groups

Covers useful lemmas for p-groups, including divisibility properties and the existence of specific elements in finite abelian groups.

Existence: Proof of Classification Theorem

Covers the proof of the Classification Theorem for finite abelian groups, demonstrating the existence of a decomposition into cyclic groups.

Group Presentations: Abelian Groups

Covers exact sequences, torsion, divisibility, and operations on abelian groups.

Symmetry and Group Theory

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

Finitely Generated Abelian Groups

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

Applications of Lagrange's Theorem

Explores the applications of Lagrange's theorem in algebra, covering corollaries, cyclic groups, and quotient groups.

Categorical Perspective: Abelian Groups

Explores abelian groups from a categorical perspective, discussing the coproduct construction.

Resoluble Groups: Quotients of Group

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

The Lemma of the Five: Abelian Groups

Presents the Lemma of the Five, a fundamental result in group theory.

Torsion and Divisibility in Group Theory

Explores the relationship between p-torsion and p-divisibility in group theory, highlighting implications of p-divisibility in exact sequences of abelian groups.