On Sylow intersections

Let G be a finite group, p a prime divisor of |G|, and T a p–subgroup of G. Define σ(T) to be the number of Sylow p–subgroups of G containing T. Call T a central p–Sylow intersection if for some Σ ⊆ Sylp (G), T = ∩(S | S є Σ), and if, in addition, T contains the center of a Sylow p–subgroup of G. This work is inspired and motivated by work of G. Stroth [J. Algebra 37 (1975), 111–120]. Generalizing an argument of his we describe finite groups in which every central p–Sylow intersection T with p–rank(T) > 2 satisfies σ(T) ≤ p.Related methods yield the description of finite groups in which every central p–Sylow intersection T with p–rank(T) ≥ 2 satisfies σ(T) ≤ 2p.