A note on the Navarro conjecture for alternating groups with abelian defect

TL;DR

This paper verifies Navarro's conjecture for p-singular characters of the principal block in alternating groups with abelian defect when p is odd, extending previous results for p=2.

Contribution

It provides the first verification of Navarro's conjecture for the case of odd primes p in alternating groups with abelian defect.

Findings

Navarro's conjecture holds for p-singular characters in the specified case

Verification extends known results from p=2 to odd primes p

Supports the broader validity of Navarro's refinement of the McKay conjecture

Abstract

In [7], G. Navarro proposed a refinement of the McKay conjecture involving a special class of Galois automorphisms. In [6] this new conjecture was verified by the author for the alternating groups A(n) when p=2. In this note the Navarro conjecture is verified for the p-singular characters of the principal block of the alternating groups in the case of abelian defect when p is odd.