The Picky and Subnormalizer Conjectures for symmetric groups

A new type of conjectures on characters of finite groups, related to the McKay conjecture, have recently been proposed. In this paper, we study these conjectures for symmetric groups.

💡 Research Summary

The paper introduces two new local‑global conjectures concerning characters of finite groups, called the Picky Conjecture (Conjecture A) and the Subnormalizer Conjecture (Conjecture B). Both are motivated by the classical McKay conjecture, which predicts an equality of the numbers of irreducible characters of degree prime to a given prime p between a group G and the normalizer of a Sylow p‑subgroup.
A p‑element x in a finite group G is called picky if it lies in a unique Sylow p‑subgroup of G. For such an element the authors define the set
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