Welcome to MathOverflow

Question

-1

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 $r+1$?

flag	asked Jan 30 at 16:44 Tom 36●1

3

What makes you think this is the case? – Steven Landsburg Jan 30 at 17:02

While Nick's answer looks reasonable (and better formulated than the question), the question itself isn't carefully stated and might better be closed. – Jim Humphreys Jan 31 at 0:10

Nick Gill · Accepted Answer · 2013-01-30 17:14:44Z UTC

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.

Question on the projective special unitary group

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.