# If p is prime, prove the group (Zp*,x) has exactly one element of order 2.

1. Nov 26, 2011

### ae1709

Hi. I need to: prove the group (Zp*,x) has exactly one element of order 2. Here, p is prime and (Zp*,x) is the set {1, 2,....., p-1} under multiplication modulo p. Any help would be much appreciated!

2. Nov 26, 2011

### Office_Shredder

Staff Emeritus
I think the easier half is: which element has order two?

It might help if you write down the condition of what it means to have order 2

3. Nov 26, 2011

### micromass

Something that might help you is knowing that $\mathbb{Z}_p^*$ is cyclic. Do you know this already??

4. Nov 26, 2011

### micromass

If you do not know the previous. Then you might want to think about the polynomial $X^2-1$ in $\mathbb{Z}_p[X]$. What do you know about its roots??