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

Showing the unit group is cyclic

  1. Oct 26, 2011 #1
    1. The problem statement, all variables and given/known data
    Let p be a positive prime and let Up be the unit group of Z/Zp. Show that Up is
    cyclic and thus Up [itex]\cong[/itex] Z/Z(p − 1).

    3. The attempt at a solution
    What do they mean by the unit group? Is that just the identity??? Is it the group [p]? I'm lost without starting the question...
  2. jcsd
  3. Oct 26, 2011 #2
    The unit group is just the group of all invertible elements (with multiplication as operation). So [itex]x\in U_p[/itex] if and only if x is invertible in [itex]\mathbb{Z}_p[/itex]. You have to show that it is cyclic (generated by 1 element).
  4. Oct 26, 2011 #3
    OK thank you!
  5. Oct 26, 2011 #4
    But now that I think of it, wouldn't Up be {e,....,p}, the entire equivalence class of Zp, since every element can be multiplied by another (its respective inverse) to get the identity?
  6. Oct 26, 2011 #5
    0 doesn't have an inverse...
  7. Oct 26, 2011 #6
    oh right, so then U would have p-1 elements. Got it!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook