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 $r+1$?
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This can be answered on a casebycase basis. Work in $SU(3,q)$ because it's easier, and observe that if $r>3$ then $r$ divides one of $q, q+1, q1, q^2q+1$. 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 $q^2q+1$, then the normalizer of a Sylow has size $3(q^2q+1)$ and its index is certainly bigger than $r+1$ for any $r$ dividing $q^2q+1$. The other cases are all similar. 

