most recent 30 from http://mathoverflow.net 2013-05-23T08:31:22Z http://mathoverflow.net/feeds/question/120337 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120337/question-on-the-projective-special-unitary-group Tom 2013-01-30T16:44:42Z 2013-01-30T17:14:44Z

<p>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 $r+1$?</p>

http://mathoverflow.net/questions/120337/question-on-the-projective-special-unitary-group/120342#120342 Nick Gill 2013-01-30T17:14:44Z 2013-01-30T17:14:44Z

<p>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.</p> <p>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.</p>