This report, titled "Infinite Abelian Groups," by Joaquin Pascual, explores the theory of infinite abelian groups, a branch of mathematics that has gained significant attention in recent years. The paper builds on the foundational work of J. Rotman's book "Theory of Groups: An Introduction." It begins by introducing the concept of infinite abelian groups and their classification into torsion groups, torsion-free groups, and divisible groups. The report delves into the properties and characteristics of these groups, including theorems that describe their structure and behavior. Key topics covered include: - The extension of groups and theorems related to direct sums and subgroups. - The properties of torsion and torsion-free groups, including theorems that show how these groups can be decomposed into simpler components. - The concept of divisibility in abelian groups and theorems that characterize divisible groups. - The structure of divisible groups, including theorems that describe their direct sums and subgroups. - The relationship between torsion and torsion-free groups, and how they can be combined to form more complex structures. The report also includes proofs of several important theorems, such as the decomposition of abelian groups into direct sums of torsion and torsion-free parts, the characterization of divisible groups, and the structure of p-primary groups. Additionally, it discusses the injective property of groups and its implications for divisibility. Overall, the report provides a comprehensive overview of the theory of infinite abelian groups, offering both theoretical insights and practical applications.This report, titled "Infinite Abelian Groups," by Joaquin Pascual, explores the theory of infinite abelian groups, a branch of mathematics that has gained significant attention in recent years. The paper builds on the foundational work of J. Rotman's book "Theory of Groups: An Introduction." It begins by introducing the concept of infinite abelian groups and their classification into torsion groups, torsion-free groups, and divisible groups. The report delves into the properties and characteristics of these groups, including theorems that describe their structure and behavior. Key topics covered include: - The extension of groups and theorems related to direct sums and subgroups. - The properties of torsion and torsion-free groups, including theorems that show how these groups can be decomposed into simpler components. - The concept of divisibility in abelian groups and theorems that characterize divisible groups. - The structure of divisible groups, including theorems that describe their direct sums and subgroups. - The relationship between torsion and torsion-free groups, and how they can be combined to form more complex structures. The report also includes proofs of several important theorems, such as the decomposition of abelian groups into direct sums of torsion and torsion-free parts, the characterization of divisible groups, and the structure of p-primary groups. Additionally, it discusses the injective property of groups and its implications for divisibility. Overall, the report provides a comprehensive overview of the theory of infinite abelian groups, offering both theoretical insights and practical applications.