Showing the unit group is cyclic

1. Oct 26, 2011

Locoism

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 $\cong$ 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. Oct 26, 2011

micromass

Staff Emeritus
The unit group is just the group of all invertible elements (with multiplication as operation). So $x\in U_p$ if and only if x is invertible in $\mathbb{Z}_p$. You have to show that it is cyclic (generated by 1 element).

3. Oct 26, 2011

Locoism

OK thank you!

4. Oct 26, 2011

Locoism

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. Oct 26, 2011

micromass

Staff Emeritus
0 doesn't have an inverse...

6. Oct 26, 2011

Locoism

oh right, so then U would have p-1 elements. Got it!