Question on the equal Sylow number in finite non-abelian simple group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:11:49Z http://mathoverflow.net/feeds/question/110315 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110315/question-on-the-equal-sylow-number-in-finite-non-abelian-simple-group Question on the equal Sylow number in finite non-abelian simple group Tom 2012-10-22T10:35:24Z 2012-10-22T13:50:05Z <p>let $G$ be a finite non-abelian simple group.If there exist $p$ and $q$ which are different prime numbers of $|G|$ such that $n_p(G)=n_q(G)$?</p> http://mathoverflow.net/questions/110315/question-on-the-equal-sylow-number-in-finite-non-abelian-simple-group/110320#110320 Answer by Peter Mueller for Question on the equal Sylow number in finite non-abelian simple group Peter Mueller 2012-10-22T11:49:53Z 2012-10-22T13:50:05Z <p>Not sure if this question really qualifies for MO.</p> <p>Anyway, the answer very much depends on the group $G$. In most cases $n_p(G)\ne n_q(G)$ for distinct prime divisors of the group order. However, there are infinitely many examples where equality occurs: If $r$ is an odd prime, then $n_p(\text{PSL}(2,r))=r(r+1)/2$ for each odd prime divisor $p$ of $r-1$.</p> <p>But there are other examples too. For instance the atlas of finite simple groups shows that in the Janko group $J_1$, the normalizers of the $3$-Sylows and $5$-Sylows have order $60$.</p>