On Minimal Degrees of Faithful Characters of Finite Groups With a T.I.Sylow p-subgroup

The finite group G is said to have T.I. Sylow p-subgroup P, if two differentconjugats of P have only the identity element in common. Using the classificationof the finite simple groups, T.R. Berger, P. Landrock and G.O. Michler provedthe following theorem in 1985, which was conjectured to hold by H.S.Leonard in1968. Theorem 1. Let G be a finite group with a T.I. Sylow p-subgroup P. If Ghas a faithful complex character X such that X(1)≤|P|~(1/2)-1,then P is normalin G.