Vanishing elements and Sylow subgroups

Abstract

Let G be a finite group and p be a prime. A vanishing conjugacy class of G is a conjugacy class of G which consists of vanishing elements of G. A p-singular vanishing conjugacy class is a vanishing conjugacy class with elements of order divisible by p. Dolfi, Pacifici, Sanus and Spiga proved that if a group has no p-singular vanishing elements, then it has a normal Sylow p-subgroup. In this note we investigate the structure of a Sylow p-subgroup P of a group G with exactly one p-singular vanishing conjugacy class. In particular, we show that if p > 3, then P′ is a subnormal subgroup of G. We also prove that |P/ O p (G)| ≤ p or G has a composition factor isomorphic to PSL2(q), where q = pf , f ≥ 2 and O p (G) denotes the largest normal p-subgroup of G.