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!

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


    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