Steinberg-like characters for finite simple groups

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Abstract Let G be a finite group and, for a prime p, let S be a Sylow p-subgroup of G. A character χ of G is called Syl p {\mathrm{Syl}_{p}} -regular if the restriction of χ to S is the character of the regular representation of S. If, in addition, χ vanishes at all elements of order divisible by p, χ is said to be Steinberg-like. For every finite simple group G, we determine all primes p for which G admits a Steinberg-like character, except for alternating groups in characteristic 2. Moreover, we determine all primes for which G has a projective FG-module of dimension | S | {\lvert S\rvert} , where F is an algebraically closed field of characteristic p.