Torsion and Divisibility

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Finitely Generated Abelian Groups

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

Group Operations and Homomorphisms in Abelian Groups

Explores operations on abelian groups, Homomorphisms, adjoints, and morphisms in group theory.

Orbit-Stabilizer Theorem: Conjugacy Classes and Direct Products

Covers the Orbit-Stabilizer theorem, conjugacy classes, and direct products in finite groups.

Categorical Perspective: Abelian Groups

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

Existence of Sylow Subgroups: Proof by Recurrence

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

Algebra: Group Actions and Direct Products

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

Symmetry and Group Theory

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

Sylow Subgroups: Group Theory

Explores the concept of p-subgroups of Sylow in group theory, emphasizing the philosophy of studying mathematical objects 'one prime at a time'.

Quotients in Abelian Groups

Covers quotients in abelian groups and the concept of free abelian groups, showing that every abelian group is isomorphic to a quotient of a free abelian group.

Applications of Lagrange's Theorem

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