Many p,q-Sylow subgroups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:15:59Z http://mathoverflow.net/feeds/question/80186 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80186/many-p-q-sylow-subgroups Many p,q-Sylow subgroups Dan Glasscock 2011-11-06T04:00:09Z 2011-11-09T12:32:32Z <p>It is a fact that the symmetric groups have as many 2-Sylow subgroups as possible. More precisely, for all $n \geq 1$, the number of 2-Sylow subgroups in $S_n$ is exactly $n!/2^{\nu_2(n!)}$, which is the index of a 2-Sylow subgroup of $S_n$. This follows from (or, depending on which direction you're coming from, proves) the fact that one (equivalently, all) 2-Sylow subgroup is self normalizing.</p> <p>It isn't too hard to show that given a prime $p$, there is a family of finite groups $(G_n)$ such that $\nu_p(|G_n|) \rightarrow \infty$ and all the $p$-Sylow subgroups of $G_n$ are self-normalizing.</p> <p>I want to generalize this to two primes in the obvious way, but I am encountering difficulty. The following would be a good start.</p> <blockquote> <p>Given distinct primes $p$, $q$, does there exist a finite group $G$ such that $pq \ \Big| \ |G|$ and all $p$-Sylow, all $q$-Sylow subgroups are self-normalizing? </p> </blockquote> http://mathoverflow.net/questions/80186/many-p-q-sylow-subgroups/80235#80235 Answer by Faisal for Many p,q-Sylow subgroups Faisal 2011-11-06T19:39:56Z 2011-11-06T19:39:56Z <p>The answer is <strong>no</strong>: see Corollary 1.3 in</p> <blockquote> <p>Robert M. Guralnick; Gunter Malle; Gabriel Navarro, <em><a href="http://www.ams.org/journals/proc/2004-132-04/S0002-9939-03-07161-2/home.html" rel="nofollow">Self-normalizing Sylow subgroups</a></em>, Proc. Amer. Math. Soc. <strong>132</strong> (2004), 973-979. </p> </blockquote>