Showing the unit group is cyclic

  • Thread starter Locoism
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  • #1
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Homework Statement


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).


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...
 

Answers and Replies

  • #2
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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).
 
  • #3
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OK thank you!
 
  • #4
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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?
 
  • #5
22,089
3,296
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?
0 doesn't have an inverse...
 
  • #6
81
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oh right, so then U would have p-1 elements. Got it!
 

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