A Characteristic Subgroup and Kernels of Brauer Characters

Note that the intersection property in Theorem A is equivalent to saying that NL(P ) is a p-group. Also, since this property is clearly independent of the choice of P in Sylp(G), it is clear that L is characteristic in G. Our interest in this characteristic subgroup was motivated by the following. Theorem B. Suppose that G is p-solvable and let L be the largest normal subgroup of G such that L∩P = L∩NG(P ), where P ∈ Sylp(G). Then L is the intersection of the kernels of the irreducible Brauer characters of G with degree not divisible by p. The assumption that G is p-solvable in Theorem B is essential. Consider, for example, the simple group G = M23 and take p = 2. Then G has a self-normalising Sylow 2-subgroup, and thus the characteristic subgroup L of Theorem A is the whole group G. But G has an irreducible Brauer character of degree 11, and hence the conclusion of Theorem B fails in this case. Received 8th June, 2005 The second author is partially supported by the Ministerio de Educación y Ciencia proyecto MTM200406067-C02-01. Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/05 $A2.00+0.00.