1. Not finding help here? Sign up for a free 30min 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!

Group Theory simple question

  1. Feb 13, 2015 #1
    1. The problem statement, all variables and given/known data
    If G is a group of even order, show that it has an element g not equal to the identity such that g^2 = 1.

    2. Relevant equations
    None

    3. The attempt at a solution
    What I wrote:

    If |G| = n, then g^n = 1 for some g in G. Thus, (g^(n/2))(g^(n/2)) = 1, so g^(n/2) is the element of order 2.

    Is this a flawed argument? there guaranteed to be an element such that g^n = 1?
     
  2. jcsd
  3. Feb 13, 2015 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The identity is, of course, its own inverse. Since the group is "of even order", i.e. it has an even number of elements, removing the identity leaves an odd number of elements. Pairing each number with its inverse, what happens?
     
  4. Feb 15, 2015 #3
    I see your arguement. Is this true though? Can't ab = 1, bc = 1, cd = 1 etc etc but ba not equal 1?
     
  5. Feb 15, 2015 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    No, it's not possible. If ab=1 then ba=1. Prove it!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Group Theory simple question
  1. Group theory question (Replies: 5)

  2. Group theory question (Replies: 3)

Loading...