Week 5 HW, Chapter 6, Problem 22, MA453, Spring 2008, Prof. Walther
Problem Statement
Prove or disprove that U(20) and U(24) are isomorphic.
Discussion
By listing out the elements of U(20) = {1,3,7,9,11,13,17,19} and then their corresponding orders which are 1,4,4,2,2,4,4,2 and then the elements of U(24) = {1,5,7,11,13,17,19,23} and their corresponding orders which are 1,2,2,2,2,2,2,2 we can see that since the largest order of any element in these groups does not agree, then U(20) and U(24) cannot be isomorphic.
-Karen Morley
That's an interesting way to explain that. It looked at it a different way. In U(20), the elements whose square is 1 are 1,9,11,19. In U(24), the elements whose square is 1 are 1,5,7,11,13,17,19,23. It follows that no isomorphism can exist because it would have to math 1-1 a set of 8 elements to one with 4. I like your explanation better though. It shows it in a simpler, easier to understand way.
-Linley Johnson
--Johns121 16:33, 10 February 2009 (UTC)
I did this similar to Karen. When my first test for isomorphism wasn't working I tried comparing the largest order of elements in U(20) and U(24) to conclude that they were not isomorphic. Glad to see someone else did something similar.
