A Special Case of the Navarro Conjecture

In [7] G.Navarro proposed a refinement of the McKay conjecture involving a special class of automorphisms. In [6] this new conjecture was verified for the alternating groups A(Π) when p = 2. In this paper, it is verified for the p-singular characters of A(Π) when |Π| = wp, w < p and p is odd. 1. McKay and Navarro conjectures Let G be a finite group, where |G| = n. Let p be a prime dividing n, D a Sylow p-group of G, and NG(D) the normalizer of D in G. Let G ∨ denote the irreducible characters of G, and Gp′ the subset of characters whose degree is relatively prime to p. The following is a well-known conjecture. Conjecture 1.1. (McKay, [1]) |Gp′| = |NG(D)p′|. Recently G. Navarro strengthened the McKay conjecture in the following way. All irreducible complex characters of G are afforded by a representation with values in the nth cyclotomic field Qn/Q (Lemma 2.15, [4]). Then the Galois group G = Gal(Qn/Q) permutes the elements of G. We denote the action of σ on χ ∈ G by χ. Then χ ∈ G is σ-fixed if its values are fixed by σ, that is, χ = χ. Let e be a nonnegative integer and consider σe ∈ G where σe(ξ) = ξpe for all p-roots of unity ξ. Define N to be the subset of G consisting of all such σe. Let Gp′,σ and NG(D) ∨ p,σ be the subsets of G ∨ p and NG(D) ∨ p respectively fixed by σ ∈ N . Conjecture 1.2. (Navarro, [7]) Let σ ∈ N . Then |Gp′,σ| = |NG(D)p′,σ|. The Navarro conjecture is equivalent to the existence of a bijection φ from Gp′ to NG(D) ∨ p that commutes with elements of N . That 2000 Mathematics Subject Classification. Primary 20C30.