Let $G$=PSU$(3,q)$ be projective special unitary group where $q$ is prime
power. I would like to know why there is not any prime $r$ such that the
number of Sylow $r$-subgroups of $G$ is $r+1$?



Question on the projective special unitary group
This can be answered on a case-by-case basis. Work in $SU(3,q)$ because it's easier, and observe that if $r>3$ then $r$ divides one of $q, q+1, q-1, q^2-q+1$. Now go through these one at a time.


E.g. if $r$ divides $q$, then $r=p$. Now either calculate the size of the normalizer of a Sylow $p$ or just observe $r+1$ is less than the minimal index of a subgroup of $G$. Similarly if $r$ divides $q^2-q+1$, then the normalizer of a Sylow has size $3(q^2-q+1)$ and its index is certainly bigger than $r+1$ for any $r$ dividing $q^2-q+1$. The other cases are all similar.


