Proving that Q/Z is an Infinite Group with Finite-Order Elements: A Comprehensive Guide

Understanding the properties of the group Q/Z is crucial for various areas of mathematics, including abstract algebra and number theory. This article provides a detailed proof on how Q/Z can be shown to be an infinite group, while every element within has finite order.

Introduction to Q/Z

Let's delve into the group mathbb{Q}/mathbb{Z}, which consists of cosets of the form q mathbb{Z}, where q in mathbb{Q}. Two rational numbers q_1 and q_2 belong to the same coset if their difference q_1 - q_2 is an integer.

Part 1: Proving Q/Z is an Infinite Group

Definition of the Group

The group mathbb{Q}/mathbb{Z} comprises cosets of the form q mathbb{Z} with q in mathbb{Q}. Two rational numbers q_1 and q_2 are in the same coset if q_1 - q_2 in mathbb{Z}.

Constructing Elements

Consider the elements of mathbb{Q}/mathbb{Z} as fractions of the form frac{m}{n} mathbb{Z}, where m and n are integers and n eq 0.

Infinitude: For each positive integer n, the fractions frac{0}{n}, frac{1}{n}, frac{2}{n}, ldots, frac{n-1}{n} are distinct in mathbb{Q}/mathbb{Z}. This means for each n, we have n distinct elements in mathbb{Q}/mathbb{Z}. Since there are infinitely many positive integers n, it follows that mathbb{Q}/mathbb{Z} is infinite.

Part 2: Proving Every Element Has Finite Order

Element Representation

Consider an arbitrary element x frac{a}{b} mathbb{Z} in mathbb{Q}/mathbb{Z}, where a and b are integers and b eq 0.

Order of an Element

The order of x is defined as the smallest positive integer k such that kx in mathbb{Z}. In other words, we need frac{ka}{b} in mathbb{Z}.

Finding the Order

We need to find the smallest k such that frac{ka}{b} is an integer. This occurs when k is a multiple of b. Specifically, if we take k b, then:

bx frac{ba}{b} mathbb{Z} a mathbb{Z} in mathbb{Z}.

Therefore, every element in mathbb{Q}/mathbb{Z} has finite order, and the order divides b.

Conclusion

Therefore, we conclude that mathbb{Q}/mathbb{Z} is an infinite group because there are infinitely many distinct cosets represented by rational numbers with different denominators. Moreover, every element in mathbb{Q}/mathbb{Z} has finite order, as each can be expressed as a fraction whose multiples yield integers.

Hence, mathbb{Q}/mathbb{Z} is an infinite group where every element has finite order.