Abstract
In this paper the following main theorem is proved. Theorem. Let G be a finite simple group with a Sylow p-normalizer in G of order 3p. If there is a nonprincipal ordinary irreducible character of degree at most 29 in the principal p-block, then G is isomorphic to one of the groups L(3, 2), A 7, U(3, 3), L(3, 3), C7(3, 4), L(3, 4), U(3, 5), or A 8.
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