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!

A cyclic group of order 15

  1. Dec 20, 2003 #1
    A cyclic group of order 15 has an element x such that the set {x^3, x^5, x^9} has exactly two elements. The number of elements in the set {x^(13n) : n is a positive integer} is :

    WHy is the answer 3? Thanks!!
  2. jcsd
  3. Dec 20, 2003 #2
    Let [tex]x[/tex] be an element of a cyclic group of order 15. If [tex]\{x^3,x^5,x^9\}[/tex] has exactly 2 elements, then one element must be the same as another. If [tex]|x|=3[/tex] then [tex]x^3=x^9[/tex]. If [tex]|x|=3[/tex] then [tex]|x^{13}|=3/gcd(3,13)}=3[/tex]. Hence [tex]|<x^{13}>|=3[/tex].

  4. Dec 20, 2003 #3
    Thanks so much for the explanation!! However I haven't actually studied group theory before so there's some things I still am not sure about. Why did you assume |x| = 3? What formula did [tex]|x^{13}|=3/gcd(3,13)}=3[/tex] come from?
    Can you give an example of a set that meets this condition?
  5. Dec 20, 2003 #4
    Your original post stated that there exists an element [tex]\inline{x}[/tex] such that [tex]\inline{\{x^3,x^5,x^9\}}[/tex] has exactly two elements. Thus I need to find two elements that are the same. There is a theorem that states if [tex]\inline{|x|=n}[/tex] then [tex]\inline{x^i=x^j}[/tex] if and only if [tex]\inline{n}[/tex] divides [tex]\inline{i-j}[/tex]. If [tex]\inline{x^3=x^5}[/tex] then [tex]\inline{|x|=2}[/tex] which is not possible in a cyclic group of order 15 because 2 does not divide 15. Likewise, [tex]\inline{x^5{\neq}x^9}[/tex]. However if [tex]\inline{|x|=3}[/tex] then [tex]\inline{x^3=x^9}[/tex] because 3 divides [tex]\inline{i-j=9-3=6}[/tex]. Also, a cyclic group of order 15 can have an element with order 3.

    There is a theorem that states if [tex]\inline{|x|=n}[/tex] then [tex]\inline{|x^k|=n/gcd(n,k)}[/tex].

    The group [tex]\inline{Z_{15}}[/tex] is the group of integers modulo 15 under addition. Both the element 5 and the element 10 have order 3.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook