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!

Two conjugate elements of a group have the same order PROOF

  1. Apr 21, 2016 #1

    RJLiberator

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Let x and y be conjugate elements of a Group G. Prove that x^n = e if and only if y^n = e, hence x and y have the same order.

    2. Relevant equations
    Conjugate elements : http://mathworld.wolfram.com/ConjugateElement.html

    3. The attempt at a solution

    Since y is a conjugate of x, there exists z ∈G such that y = (zxz^-1).

    If x^n = e, then y^n = (zxz^-1)^n = (zx^nz^-1) = (zez^-1) = zz^-1 = e.

    Similiarly, if y^n = e, then e=y^n = (zxz^-1)^n = (zx^nz^-1). Multiplying on the left by z^-1 and on the right by z we see
    z^-1ez = z^-1zx^nz^-1z and so e = ex^ne = x^n

    Done.


    My concern is in the definition of conjugate elements. If x and y are conjugate elements of a group G, does that necessarily mean y is a conjugate of x?

    I've supplied the definition of conjugate elements in the 'relevant equations' part of this thread.
     
  2. jcsd
  3. Apr 21, 2016 #2

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I don't like that definition because it never clearly states the necessary and sufficient conditions for being conjugate. It also blurs the distinction between the - at first - very different statements 'a and b are conjugate' and 'a is conjugate to b'. Use this definition instead:

    If a and b are elements of group G then we say that a is conjugate to b iff there exists ##g\in G## such that ##gag^{-1}=b##.

    It is then simple to prove that a is conjugate to b iff b is conjugate to a.

    Hence, we can refer to two elements as being conjugate, without specifying an order, ie the statement 'a and b are conjugate in G' is well-defined, and means 'a is conjugate to b', which is the same as 'b is conjugate to a'.
     
  4. Apr 21, 2016 #3

    RJLiberator

    User Avatar
    Gold Member

    So, when the question states "conjugate elements" we can think of the definition as

    The way the question is worded, and by this definition, then I can safely assume that there exists z ∈G such that y = (zxz^-1) and my proof follows.
     
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: Two conjugate elements of a group have the same order PROOF
Loading...