I have two sets with $n>2$ natural number:
$S=\{1\le a \le n:(a,n)=1,a^{n-1}\not\equiv 1\pmod n\}$
$T=\{1\le b \le n:(b,n)=1,b^{n-1}\equiv 1\pmod n\}$
Can anyone explain me if there are prime numbers $n$ for whom $S\neq\emptyset$?and are there any composite numbers $n$ for whom $S=\emptyset$? I think that i have to apply an primality test but i do not know what. Also how i can prove this: Also how to prove: if $S\neq\emptyset$ show that $|S|\geφ(n)/2$ without knowing anything for $φ(n)$