Sum of the squares of the $p'$-character degrees

TL;DR

This paper investigates the sum of squares of irreducible character degrees not divisible by a prime p, relating it to p-Sylow normalizers, and proves a conjecture for p=2 and other cases.

Contribution

It establishes a relationship between character degree sums and p-Sylow normalizers, proving a conjecture for p=2 and additional cases.

Findings

Proved the conjecture for p=2.

Analyzed the sum of character degrees not divisible by p.

Connected character degree sums with p-Sylow normalizers.

Abstract

We study the sum of the squares of the irreducible character degrees not divisible by some prime $p$, and its relationship with the the corresponding quantity in a $p$-Sylow normalizer. This leads to study a recent conjecture by E. Giannelli, which we prove for $p=2$ and in some other cases.