# Showing the unit group is cyclic

• Locoism

## 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 $\cong$ 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...

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).

OK thank you!

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?

Locoism said:
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...

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