Let G be an Abelian group and let H be the subgroup...

Let $G$ be an Abelian group and let $H$ be the subgroup consisting of all elements of $G$ that have finite order. (See Exercise 20 in the Supplementary Exercises for Chapters $1-4 .$ ) Prove that every nonidentity element in $G / H$ has infinite order.

Key Concepts

Abelian Groups

An Abelian group is a group in which the operation is commutative, meaning the order in which two elements are combined does not affect the result. This property simplifies the structure and analysis of subgroups, particularly when exploring the behavior of elements with respect to their order.

Subgroups

A subgroup is a subset of a group that forms a group under the same operation. In the context of the problem, a crucial subgroup is the set of elements with finite order, also known as the torsion subgroup, which is used to partition the original group into cosets.

Element Order (Finite and Infinite)

The order of an element in a group is defined as the smallest positive integer for which some power of the element equals the identity. Elements that do not satisfy this condition for any finite integer have infinite order. Distinguishing between finite and infinite order is essential for understanding the structure of a quotient group in the given problem.

Torsion Subgroup

The torsion subgroup consists of all elements in a group that have finite order. In Abelian groups, this subgroup plays a fundamental role in decomposing the group into simpler components and is critical when considering how elements behave in the quotient by this subgroup.

Quotient Groups

A quotient group is created by partitioning a group into disjoint subsets called cosets with respect to a normal subgroup. The structure of a quotient group reflects the properties of the original group modded out by the subgroup, and in this problem, it is used to show that nonidentity elements of the quotient have infinite order.

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