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Character Values and Dade's Conjecture

Let G be an arbitrary finite group and fix a prime p. Let S be a Sylow p-subgroup of G. In [19], J. McKay proposed the following conjecture (McKay conjecture) : The numbers of irreducible characters of G and NG(S ) having degrees not divisible by p are equal when p = 2. In [1], J. L. Alperin generalized this conjecture for arbitrary primes and p-blocks of G (Alperin-McKay conjecture). Recently, I. M. Isaacs and G. Navarro extended it including the p′-parts of character degrees in [15]. Moreover in [21], Navarro proposed a strong form of the McKay conjecture, which concerns character values. In [2], Alperin stated the weight conjecture, which described the number of irreducible modular characters by p-local information. In [16], R. Knörr and G. Robinson restated this conjecture using the alternating sum of the numbers of characters. E. C. Dade proposed an extention of this conjecture concerning the defect of characters in [10], and there are sevral forms of Dade’s conjecture. Note that Dade’s conjecture (the projective form) implies the Alperin-McKay conjecture. (See [11].) The extended McKay conjecture by Isaacs and Navarro was subdivided into Dade type by K. Uno in [23]. In this paper, moreover, we subdivide the strong form of the McKay conjecture by Navarro into Dade type, and we prove this extended conjecture (Conjecture 1.4) for several sporadic simple groups (Theorem 1.5). For the calculations the GAP system is used. Now we mention some notations used in this paper. We denote by N : G = NoG and N.G a split and non-split extention of N by G, respectively. We use n to denote a group of order n, and Zn to denote a cyclic group of order n. For a prime p, an elementary abelian group of order pn is denoted by Epn or simply pn. We denote by p + an extraspecial p-group of order p1+2γ and type +. We use Q8 to denote the quaternion group, and D2n to denote the dihedral group of order 2n. We use Sn and An to denote the symmetric group and the alternating group of degree n, respectively. We denote by Fm n a Frobenius group with kernel Zn and complement Zm. For a subgroup H of G, we use H′ to denote a subgroup of G such that H H′ but H′ is not G-conjugate to H. Other notations including those for sporadic simple groups are taken from Atlas [9]. Throughout the paper, a character means an irreducible complex character. Let G be a finite group and H a subgroup of G. When a prime p is fixed, for a positive integer n, we denote by np the p-part of n, and by np′ the p′-part of n. We denote by Bl(H) the set of all p-blocks of H. The principal block of H is denoted by