Hallo,

Me again. Here's the last of four questions I have for today. (I'll probably asking some questions about Set Theory soon, but not today).

Again, this is a homework problem, so please, only hints

Problem:

(a) Prove that if $\displaystyle n$ is even, then exactly one nonidentity element of $\displaystyle \mathbb{Z}_n$ is its own inverse.

(b) Prove that if $\displaystyle n$ is odd, then no nonidentity element of $\displaystyle \mathbb{Z}_n$ is its own inverse.

(c) Prove that $\displaystyle [0] \oplus [1] \oplus \cdots \oplus [n - 1]$ equals either $\displaystyle [0]$ or $\displaystyle [n/2]$ in $\displaystyle \mathbb{Z}_n$

(d) What does part (c) imply about $\displaystyle 0 + 1 + \cdots + (n - 1)$ modulo $\displaystyle n$?

Things that might come in handy:

Nothing really. Again, anyone who can help me will already know all they need to know by looking at the symbols and interpreting them.

What I've tried:

I honestly have not tried anything as yet. I don't think this is a particularly hard question. I remember being confused about something when I read it through the first time though, so I post it just in case, so as to not leave it for the last minute (which is when I usually do my homework).

I won't look at any hints given here until I've tried to do the problem myself (you can trust me ). So feel free to respond anyway.

Thanks guys and gals!