The Isaacs–Navarro conjecture for the alternating groups

Abstract

A recent refinement of the McKay conjecture is verified for the case of the alternating groups. The argument builds upon the verification of the conjecture for the symmetric groups [P. Fong, The Isaacs-Navarro conjecture for symmetric groups, J. Algebra 250 (1) (2003) 154-161].

The Isaacs–Navarro conjecture for covering groups of the symmetric and alternating groups in odd characteristic

“…For any n ∈ N, we write n = n p n p ′ , with n p = p ν (n) Isaacs and Navarro proved Conjecture 1.1 whenever D is cyclic, or G is p-solvable or sporadic. P. Fong proved it for symmetric groups S(n) in [2], and R. Nath for alternating groups A(n) in [8]. In this paper, we prove that Conjecture 1.1 holds in all the covering groups of the symmetric and alternating groups, provided p is odd (Theorem 5.1).…”

Local-global conjectures in the representation theory of finite groups

“…(Note that this is a true refinement of Conjecture 2.2 whenever p ≥ 5.) This has been shown to hold for example for S n , A n and their double covers by Fong [37], Nath [79] and Gramain [42] respectively. Two further refinements on the properties of the required bijection concerning the action of those Galois automorphisms fixing a prime ideal above p were put forward in the same paper [48], and by Navarro [73] respectively.…”

“…Nath [79] and Gramain [42] respectively. Two further refinements on the properties of the required bijection concerning the action of those Galois automorphisms fixing a prime ideal above p were put forward in the same paper [48], and by Navarro [73] respectively.…”

Local-Global Conjectures and Blocks of Simple Groups

“…Alexandre Turull [83] showed that it holds for all solvable groups. This was shown to hold for alternating groups by Nath [65] Another such refinement, which is currently being studied quite intensely, was proposed by Navarro [66]; it proposes that the bijection Ω should also be equivariant with respect to Gal(Q p /Q p ); see Brunat and Nath [16] for the case of alternating groups, and Ruhstorfer [74] for groups of Lie type in their defining characteristic. where P O is the maximal ideal containing p. Richard Brauer showed how to associate to any p-block B of G a p-subgroup D ≤ G of G, unique up to conjugation, called defect group of B.…”