Question on the projective special unitary group - MathOverflow 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 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 Answer by Nick Gill for Question on the projective special unitary group 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>