Problem 6 - A Book of Abstract Algebra

Question

Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove the following: In \(\mathbb{Q} / \mathbb{Z}\), every element has finite order.

Step-by-Step Solution

Answer

Every element of \( \mathbb{Q} / \mathbb{Z} \) has finite order because any rational number can be represented with a denominator that becomes its order.

Step 1: Understanding the Setting

We need to demonstrate that each element of the quotient group \( \mathbb{Q} / \mathbb{Z} \) has finite order. Here, \( \mathbb{Q} \) represents the group of rational numbers under addition, and \( \mathbb{Z} \) is embedded as a subgroup of \( \mathbb{Q} \). The quotient group \( \mathbb{Q} / \mathbb{Z} \) consists of cosets of \( \mathbb{Z} \).

Step 2: Identifying Elements of the Group

An element of \( \mathbb{Q} / \mathbb{Z} \) can be represented as an equivalence class \( q + \mathbb{Z} \), where \( q \in \mathbb{Q} \). This class consists of elements \( q + z \) for all \( z \in \mathbb{Z} \).

Step 3: Determining Element Order

To find the order of an element \( q + \mathbb{Z} \) in \( \mathbb{Q} / \mathbb{Z} \), we seek the smallest positive integer \( n \) such that \( n(q + \mathbb{Z}) = 0 + \mathbb{Z} \). This translates to finding \( n \) such that \( nq \in \mathbb{Z} \).

Step 4: Expressing as a Rational Number

Given \( q \in \mathbb{Q} \), express \( q = \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). To ensure \( nq \) becomes an integer, choose \( n = b \). Thus, \( nq = \frac{a}{b} \times b = a \), which is an integer.

Step 5: Verifying Finite Order

With \( n = b \), the element satisfies \( n(q + \mathbb{Z}) = a + \mathbb{Z} = 0 + \mathbb{Z} \), indicating that \( q + \mathbb{Z} \) indeed has finite order \( n \). Thus, every element of \( \mathbb{Q} / \mathbb{Z} \) has finite order.

Key Concepts

Quotient Group

A quotient group, sometimes referred to as a factor group, is a unique type of group that results from dividing one group by a normal subgroup. In our exercise, we consider the quotient group \( \mathbb{Q} / \mathbb{Z} \) where \( \mathbb{Q} \) is the group of rational numbers and \( \mathbb{Z} \) is the subgroup of integers.

The process of forming a quotient group involves grouping together elements of a larger group \( G \) into cosets of a normal subgroup \( H \). Each element of a quotient group is represented as a coset, like \( q + \mathbb{Z} \) in our example.

Quotient groups are a core concept in group theory as they help in analyzing the structure and properties of groups by simplifying complex groups into simpler, more manageable pieces.

Normal Subgroup

A normal subgroup is a crucial concept for forming quotient groups. A subgroup \( H \) of a group \( G \) is considered normal if it satisfies the condition that for every element \( g \in G \), the left coset \( gH \) is equal to the right coset \( Hg \).

For the subgroup \( \mathbb{Z} \) in the group of rational numbers \( \mathbb{Q} \), this means \( \mathbb{Z} \) must fit "nicely" within \( \mathbb{Q} \) so our operations with the quotient are well-defined and independent of the representatives chosen.

Normal subgroups are foundational in group theory because they allow the construction of quotient groups, essentially "factoring" the original group \( G \) using \( H \). This helps in understanding the larger group's properties by examining these smaller, derived groups.

Order of an Element

The order of an element in a group is the smallest positive integer \( n \) such that \( n \) times the element gives the identity element of the group.

In the context of \( \mathbb{Q} / \mathbb{Z} \), the order of an element \( q + \mathbb{Z} \) is determined by finding such \( n \) where multiplying \( q \) by \( n \) results in an integer. The identity element here is \( 0 + \mathbb{Z} \).

Understanding the order helps in dissecting group structures and is pivotal for proving properties like an element's finite order in quotient groups, directly relating to the structure of \( \mathbb{Q} / \mathbb{Z} \).

Rational Numbers

Rational numbers, denoted by \( \mathbb{Q} \), are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). They form a group under the operation of addition.

In the exercise, we look at rational numbers from the perspective of group theory, particularly how they interact with integers when forming the quotient group \( \mathbb{Q} / \mathbb{Z} \).

Using rational numbers in this context allows us to explore their properties within a more abstract mathematical framework, such as understanding how they can form cycles or have properties like finite order when considered modulo \( \mathbb{Z} \).

Integers

Integers, represented by \( \mathbb{Z} \), are whole numbers that can be positive, negative, or zero. They are foundational in number theory and naturally form a subgroup of rational numbers \( \mathbb{Q} \), specifically in the context of group theory as seen here.

In our example, \( \mathbb{Z} \) acts as a normal subgroup of \( \mathbb{Q} \), enabling the formation of the quotient group \( \mathbb{Q} / \mathbb{Z} \). Here, each coset is formed by adding an integer to a fixed rational number, capturing the structure defined by integers within rational numbers.

This interaction demonstrates the integral role integers play within more complex group structures and how they serve as building blocks for constructing quotient groups and examining their properties, such as the finite order of elements.