Sylow intersections and Frobenius ratios

Let G be a finite group and p a prime dividing its order | G |, with p -part |G|ₚ | G | p , and let Gₚ G p denote the set of all p -elements in G . A well known theorem of Frobenius tells us that fₚ(G)=|Gₚ|/|G|ₚ f p ( G ) = | G p | / | G | p is an integer. As Gₚ G p is the union of the Sylow p -subgroups of G , this Frobenius ratio fₚ(G) f p ( G ) evidently depends on the number sₚ(G)=|Sylₚ(G)| s p ( G ) = | Syl p ( G ) | of Sylow p -subgroups of G and on Sylow intersections . One knows that sₚ(G)=1+kp s p ( G ) = 1 + k p and fₚ(G)=1+ (p-1) f p ( G ) = 1 + ℓ ( p - 1 ) for nonnegative integers k, k , ℓ , and that fₚ(G)<sₚ(G) f p ( G ) < s p ( G ) unless G has a normal Sylow p -subgroup. In order to get lower bounds for fₚ(G) f p ( G ) we, study the permutation character π =π ₚ(G) π = π p ( G ) of G in its transitive action on Sylₚ(G) Syl p ( G ) via conjugation (Sylow character). We will get, in particular, that fₚ(G)≥ sₚ(G)/rₚ(G) f p ( G ) ≥ s p ( G ) / r p ( G ) where rₚ(G) r p ( G ) denotes the number of P -orbits on Sylₚ(G) Syl p ( G ) for any fixed P∈ Sylₚ(G) P ∈ Syl p ( G ) . One can have ≥ k≥ 1 ℓ ≥ k ≥ 1 only when P is irredundant for Gₚ G p , that is, when P is not contained in the union of the Q≠ P Q ≠ P in Sylₚ(G) Syl p ( G ) and so P= Q≠ P(P∩ Q) P ^ = ⋃ Q ≠ P ( P ∩ Q ) a proper subset of P . We prove that ≥ k ℓ ≥ k when |P|≤ |P|/p | P ^ | ≤ | P | / p .