Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Prove that the set of all rational numbers of the form m/n, where m, n E Z and n is square- free, is a subgroup of Q (under +).

Prove that the set of all rational numbers of the form m/n, where m, n E Z and n is square- free, is a subgroup of Q (under +).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
100%
**Problem Statement:** Prove that the set of all rational numbers of the form \( m/n \), where \( m, n \in \mathbb{Z} \) and \( n \) is square-free, is a subgroup of \( \mathbb{Q} \) (under \( + \)). **Explanation:** The problem asks us to demonstrate that a specific subset of rational numbers forms a subgroup. In group theory, a subgroup must satisfy three criteria under its operation (in this case, addition): 1. **Closure:** If \( a \) and \( b \) are elements of the set, then \( a + b \) must also be an element of the set. 2. **Identity:** The set must contain the identity element of the larger group. For rational numbers (under addition), the identity element is 0. 3. **Inverses:** For every element \( a \) in the set, there must be an element \( b \) in the set such that \( a + b = 0 \). The challenge is to show that rational numbers of the form \( m/n \), where \( n \) is square-free (a number not divisible by any perfect square other than 1), satisfy these subgroup criteria within \( \mathbb{Q} \).
Transcribed Image Text:**Problem Statement:** Prove that the set of all rational numbers of the form \( m/n \), where \( m, n \in \mathbb{Z} \) and \( n \) is square-free, is a subgroup of \( \mathbb{Q} \) (under \( + \)). **Explanation:** The problem asks us to demonstrate that a specific subset of rational numbers forms a subgroup. In group theory, a subgroup must satisfy three criteria under its operation (in this case, addition): 1. **Closure:** If \( a \) and \( b \) are elements of the set, then \( a + b \) must also be an element of the set. 2. **Identity:** The set must contain the identity element of the larger group. For rational numbers (under addition), the identity element is 0. 3. **Inverses:** For every element \( a \) in the set, there must be an element \( b \) in the set such that \( a + b = 0 \). The challenge is to show that rational numbers of the form \( m/n \), where \( n \) is square-free (a number not divisible by any perfect square other than 1), satisfy these subgroup criteria within \( \mathbb{Q} \).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS