Sylow intersections and Frobenius ratios

Abstract

Let G be a finite group and p a prime dividing its order |G|, with p-part \(|G|_p\), and let \(G_p\) denote the set of all p-elements in G. A well known theorem of Frobenius tells us that \(f_p(G)=|G_p|/|G|_p\) is an integer. As \(G_p\) is the union of the Sylow p-subgroups of G, this Frobenius ratio \(f_p(G)\) evidently depends on the number \(s_p(G)=|\textrm{Syl}_p(G)|\) of Sylow p-subgroups of G and on Sylow intersections. One knows that \(s_p(G)=1+kp\) and \(f_p(G)=1+\ell (p-1)\) for nonnegative integers \(k, \ell \), and that \(f_p(G)<s_p(G)\) unless G has a normal Sylow p-subgroup. In order to get lower bounds for \(f_p(G)\) we, study the permutation character \({\pi }={\pi }_p(G)\) of G in its transitive action on \(\textrm{Syl}_p(G)\) via conjugation (Sylow character). We will get, in particular, that \(f_p(G)\ge s_p(G)/r_p(G)\) where \(r_p(G)\) denotes the number of P-orbits on \(\textrm{Syl}_p(G)\) for any fixed \(P\in \textrm{Syl}_p(G)\). One can have \(\ell \ge k\ge 1\) only when P is irredundant for \(G_p\), that is, when P is not contained in the union of the \(Q\ne P\) in \(\textrm{Syl}_p(G)\) and so \(\widehat{P}=\bigcup _{Q\ne P}(P\cap Q)\) a proper subset of P. We prove that \(\ell \ge k\) when \(|\widehat{P}|\le |P|/p\).