1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Klein 4-group

  1. Dec 8, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that Klein 4 group is not isomorphic with ##Z_4##.


    2. Relevant equations
    Klein group has four elements ##\{e,a,b,c\}## such that ##e^2=e,a^2=e,b^2=e,c^2=e##
    As far as I know ##Z_4## group is ##(\{\pm 1,\pm i\},\cdot)##. Right?


    3. The attempt at a solution
    As far as I know I can say group ##Z_4## is cyclic (all elements I could get as ##i^n,n=1,2,3,4##) and group and Klein group is not.
    Q.E.D.
    Is this correct prove?
    Klein group has four element of order ##2##, and ##Z_4## group has one element of order ##4##, two element of order ##2## and one element of order one. Right?
     
  2. jcsd
  3. Dec 8, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, that's a good proof. But you've got some problems with counting orders. ##e^1## is also equal to ##e##. Go back and count them carefully and say which elements have which orders.
     
  4. Dec 8, 2013 #3
    So ##i^4=1=e## has order ##1##. ##i^2=-1## has order 2. ##i^3=-i## has order ##4## and ##i## has order ##4##.
     
  5. Dec 8, 2013 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, that's better. And the Klein group has 3 elements of order 2, and 1 element of order 1, yes?
     
  6. Dec 8, 2013 #5
    Yes! Thanks!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted