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Using the classification of the finite simple groups we show in this article that a faithful complex character X of a finite group G with a nonnormal T.I. Sylow p-subgroup P has degree x(l) > VP1 1. This result verifies a conjecture of H. S. Leonard [10]. Introduction. Let p be a fixed prime, and let G be a finite group with a T.I. Sylow p-subgroup P. That is, two different conjugates of P have only the identity element in common. In [10] H. S. Leonard conjectured that if G has a faithful complex character X with degree x(1) 1, and containing a strongly p-embedded subgroup. If p = 2, then Proposition 2.3 is only a restatement of Suzuki's theorem [13]. After these preparations Leonard's conjecture is proved in ?3. In Remark 3.3 we show that the bound of Theorem 3.2 cannot be replaced by P1), which is the bound of Sibley's theorem [12]. For notation and terminology we refer to the books by Feit [1], Gorenstein [2,3], Huppert [5], Huppert and Blackburn [6], and Landrock [9]. All character tables of finite simple groups used here are contained in the CAS-system [11] of J. Neubuser, H. Pahlings, and W. Plesken (TH. Aachen, Federal Republic of Germany). 1. Reduction to almost simple groups. In this section we determine the structure of a finite group G of minimal order among the groups H without a normal Sylow p-subgroup, but satisfying the hypothesis of Leonard's conjecture. The following lemma due to Feit [1, p. 123] is our basic tool. Received by the editors March 10, 1985 and, in revised form, August 15, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 20C05, 20C10, 20C11, 20C15, 20C20, 20F26; Secondary 20D05, 20E32, 20D20. ?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

Minimal degrees of faithful characters of finite groups with a T.I. Sylow 𝑝-subgroup