Hi !

I've just learnt today that I studied a big part of group theory last year

So I have two questions that I knew the answer, but forgot it...

1/ Is the order of an element in $\displaystyle \mathbb{Z}/n \mathbb{Z}$ an integer between m : $\displaystyle 1 \leqslant m \leqslant n$, or $\displaystyle 1 < m < n$, or $\displaystyle 1 \leqslant m \leqslant \varphi(n)$, or $\displaystyle 1<m \leqslant \varphi(n)$ ?

2/ How to prove that if $\displaystyle a \wedge n=1$, there is a unique $\displaystyle m \leqslant n$ (or <n, depending on the answer of the first question :P) such that $\displaystyle a^m \equiv 1 (\bmod n)$ ?

And here is an extra question :

3/ Basically, what's the difference between Carmichael's indicator and Euler's totient function ?

Thanks