Torsion group

Torsion group in the context of abstract algebra is a fundamental concept that plays a crucial role in the study of group theory, a branch of mathematics that deals with the algebraic structures known as groups. A torsion group is specifically related to the concept of torsion elements within a group. This article aims to elucidate the definition, properties, and examples of torsion groups, providing a comprehensive understanding of their significance in mathematical theory and applications.

Definition

A torsion group, also known as a periodic group, is a group in which every element has finite order. The order of an element g in a group G is the smallest positive integer n such that gⁿ = e, where e is the identity element of the group. If no such n exists, the element is said to have infinite order. In a torsion group, all elements except the identity element have a finite order.

Formal Definition

Given a group G, it is called a torsion group if for every element g in G, there exists a positive integer n such that gⁿ = e, where e is the identity element of G.

Examples

1. The cyclic group Z_n, which consists of the integers modulo n under addition, is a torsion group because every element x in Z_n satisfies x + n = x, indicating that every element has finite order n. 2. The group of roots of unity, which consists of all complex numbers z such that zⁿ = 1 for some positive integer n, is a torsion group. Each element in this group has an order that divides n. 3. Finite groups, by definition, are torsion groups since the order of the group (and thus the order of any element within the group) is finite.

Properties

1. Closure under taking subgroups and quotients: If G is a torsion group, then any subgroup of G is also a torsion group. Similarly, any quotient group of G is a torsion group. 2. Direct products: The direct product of two torsion groups is also a torsion group. 3. Torsion-free groups: A group is called torsion-free if the only torsion element it contains is the identity element. The concept of torsion-free groups is the opposite of torsion groups.

Applications

Torsion groups find applications in various areas of mathematics, including algebraic topology, algebraic geometry, and number theory. For example, in algebraic topology, the study of torsion subgroups of the homology groups of a topological space can provide important information about the space's structure.

See Also

• Group theory
• Cyclic group
• Roots of unity
• Finite group
• Torsion subgroup