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Homework Help: 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!
     
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