Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Group Theory, cyclic group proof

  1. Jun 1, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that Z sub n is cyclic. (I can't find the subscript, but it should be the set of all integers, subscript n.)


    2. Relevant equations


    Let (G,*) be a group. A group G is cyclic if there exists an element x in G such that G = {(x^n); n exists in Z.}

    (Z is the set of all integers)

    3. The attempt at a solution

    * is a binary operation, and for my purposes, is either additive (+) or multiplicative (x).

    Multiplicative does not work because the multiplicative inverse of, say, 2 is not an integer. So the operation must be additive. So I can rewrite the equation for (G,+) as:

    G = {nx; n exists in Z}

    but that's where I get stuck. Thanks for the help!!
     
    Last edited: Jun 1, 2010
  2. jcsd
  3. Jun 1, 2010 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    How about taking x = 1 as your generator?
     
  4. Jun 1, 2010 #3
    Every cyclic group has a generator.

    What is your generator in this case?

    edit: nm already beaten too it
     
  5. Jun 1, 2010 #4
    thanks to both!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook