Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(a) Let G be a group and let x E G. Suppose that (x) is an infinite cyclic group. Let m and n be positive integers. We know that (x) and (r") are subgroups of (x), hence their intersection (m) n (rn) is a subgroup of (r). As proved in class, any subgroup of a cyclic group is cyclic, so (xm) n (an) is cyclic. Find a generator. (It will be xP for some choice of p. Your job is to find p.) Justify your answer. (It may help to work through some examples as a warm-up.)

(a) Let G be a group and let x E G. Suppose that (x) is an infinite cyclic group. Let m and n be positive integers. We know that (x) and (r") are subgroups of (x), hence their intersection (m) n (rn) is a subgroup of (r). As proved in class, any subgroup of a cyclic group is cyclic, so (xm) n (an) is cyclic. Find a generator. (It will be xP for some choice of p. Your job is to find p.) Justify your answer. (It may help to work through some examples as a warm-up.)

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
I need help with 6a
6. (a) Let \( G \) be a group and let \( x \in G \). Suppose that \( \langle x \rangle \) is an infinite cyclic group. Let \( m \) and \( n \) be positive integers. We know that \( \langle x^m \rangle \) and \( \langle x^n \rangle \) are subgroups of \( \langle x \rangle \), hence their intersection \( \langle x^m \rangle \cap \langle x^n \rangle \) is a subgroup of \( \langle x \rangle \). As proved in class, any subgroup of a cyclic group is cyclic, so \( \langle x^m \rangle \cap \langle x^n \rangle \) is cyclic. Find a generator. (It will be \( x^p \) for some choice of \( p \). Your job is to find \( p \).) Justify your answer. (It may help to work through some examples as a warm-up.) (b) Let \( G \) be a group and let \( x \in G \). Suppose that \( \langle x \rangle \) is a cyclic group of order \( N \). Let \( m \) and \( n \) be positive integers. We know that \( \langle x^m \rangle \) and \( \langle x^n \rangle \) are subgroups of \( \langle x \rangle \), hence their intersection \( \langle x^m \rangle \cap \langle x^n \rangle \) is a subgroup of \( \langle x \rangle \). As proved in class, any subgroup of a cyclic group is cyclic, so \( \langle x^m \rangle \cap \langle x^n \rangle \) is cyclic. Find a generator. (It will be \( x^p \) for some choice of \( p \). Your job is to find \( p \).) Justify your answer. (It may help to work through some examples as a warm-up. Note that we are not assuming that \( m \) and \( n \) are divisors of \( N \).)
Transcribed Image Text:6. (a) Let \( G \) be a group and let \( x \in G \). Suppose that \( \langle x \rangle \) is an infinite cyclic group. Let \( m \) and \( n \) be positive integers. We know that \( \langle x^m \rangle \) and \( \langle x^n \rangle \) are subgroups of \( \langle x \rangle \), hence their intersection \( \langle x^m \rangle \cap \langle x^n \rangle \) is a subgroup of \( \langle x \rangle \). As proved in class, any subgroup of a cyclic group is cyclic, so \( \langle x^m \rangle \cap \langle x^n \rangle \) is cyclic. Find a generator. (It will be \( x^p \) for some choice of \( p \). Your job is to find \( p \).) Justify your answer. (It may help to work through some examples as a warm-up.) (b) Let \( G \) be a group and let \( x \in G \). Suppose that \( \langle x \rangle \) is a cyclic group of order \( N \). Let \( m \) and \( n \) be positive integers. We know that \( \langle x^m \rangle \) and \( \langle x^n \rangle \) are subgroups of \( \langle x \rangle \), hence their intersection \( \langle x^m \rangle \cap \langle x^n \rangle \) is a subgroup of \( \langle x \rangle \). As proved in class, any subgroup of a cyclic group is cyclic, so \( \langle x^m \rangle \cap \langle x^n \rangle \) is cyclic. Find a generator. (It will be \( x^p \) for some choice of \( p \). Your job is to find \( p \).) Justify your answer. (It may help to work through some examples as a warm-up. Note that we are not assuming that \( m \) and \( n \) are divisors of \( N \).)
Expert Solution
steps

Step by step

Solved in 4 steps with 4 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