My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
February 10th, 2018, 01:41 AM   #1
Member
 
Joined: Sep 2011

Posts: 97
Thanks: 1

Isomorphism problem

Hi, I have attached the question and the solutions to part a and b of this question. Would like someone to verify if I have done anything wrong. Greatly appreciate it! Thanks.

Would also like to check if there is a simpler method to prove f is an isomorphism? Thanks
Attached Images
File Type: jpg q4.jpg (14.6 KB, 4 views)
File Type: jpg Webp.net-resizeimage.jpg (83.6 KB, 9 views)
Alexis87 is offline  
 
February 10th, 2018, 07:30 AM   #2
Senior Member
 
Joined: Aug 2017
From: United Kingdom

Posts: 169
Thanks: 51

Math Focus: Algebraic Number Theory, Arithmetic Geometry
In both parts, you've started your argument by assuming what you're trying to prove.

In (a), you're being asked to prove that $[a]_6 = [b]_6 \implies ([a]_2, [a]_3) = ([b]_2, [b]_3)$, but you start by assuming this is true in the first place! What you could instead do is start with the idea in your second line to get something like:

If $[a]_6 = [b]_6$ then $a-b = 6t = 2 \times 3t$ for some integer $t$. So $2$ divides $a-b$ (i.e. $[a]_2 = [b]_2$) and $3$ divides $a-b$ (i.e. $[a]_3 = [b]_3$). Hence $f([a]_6) = ([a]_2, [a]_3) = ([b]_2, [b]_3) = f([b]_6)$.

In part (b), you've got the same issue. You're being asked to prove $f$ is an isomorphism, but you start by assuming it's an isomorphism! It seems like you've done most of the work needed for the proof (computing $f([a]_6$ for each $a$, showing $f$ respects addition, though you are missing an argument to show it respects multiplication) but you've got your argument back to front. You might like to have another attempt, bearing all this in mind.
cjem is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
isomorphism, problem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Complexity problem regarding Graph Isomorphism. jim198810 Computer Science 0 May 9th, 2015 03:47 AM
Isomorphism mathbalarka Abstract Algebra 2 November 4th, 2012 10:53 PM
Isomorphism bewade123 Abstract Algebra 2 February 14th, 2012 04:12 PM
Isomorphism mia6 Linear Algebra 1 November 10th, 2010 08:31 AM
isomorphism problem Ujjwal Linear Algebra 1 November 8th, 2008 05:48 PM





Copyright © 2018 My Math Forum. All rights reserved.