This paper presents the most important results in infinite abelian groups, following the exposition in J. Rotman's book, "Theory of Groups: An Introduction." It also includes some exercises from Rotman. The study of infinite abelian groups is reduced to the study of torsion groups, torsion-free groups, and an extension problem. Another classification reduces the study to divisible and reduced groups. The paper discusses free abelian groups, the basis and fundamental theorems of finitely generated abelian groups, and torsion and torsion-free groups of rank 1. It assumes familiarity with elementary group theory and finite abelian groups, and applies Zorn's lemma and results from vector spaces. The paper includes several theorems and definitions, such as the torsion subgroup, direct sum and product of groups, and the structure of infinite abelian groups. It also discusses divisible groups, their properties, and the structure of the group Q/Z. The paper proves that every abelian group is an extension of a torsion group by a torsion-free group, and that every torsion group is the direct sum of p-primary groups. It also shows that the group Q/Z is isomorphic to the direct sum of its p-primary components. The paper concludes with theorems on the structure of divisible groups and the injective property of divisible groups.This paper presents the most important results in infinite abelian groups, following the exposition in J. Rotman's book, "Theory of Groups: An Introduction." It also includes some exercises from Rotman. The study of infinite abelian groups is reduced to the study of torsion groups, torsion-free groups, and an extension problem. Another classification reduces the study to divisible and reduced groups. The paper discusses free abelian groups, the basis and fundamental theorems of finitely generated abelian groups, and torsion and torsion-free groups of rank 1. It assumes familiarity with elementary group theory and finite abelian groups, and applies Zorn's lemma and results from vector spaces. The paper includes several theorems and definitions, such as the torsion subgroup, direct sum and product of groups, and the structure of infinite abelian groups. It also discusses divisible groups, their properties, and the structure of the group Q/Z. The paper proves that every abelian group is an extension of a torsion group by a torsion-free group, and that every torsion group is the direct sum of p-primary groups. It also shows that the group Q/Z is isomorphic to the direct sum of its p-primary components. The paper concludes with theorems on the structure of divisible groups and the injective property of divisible groups.