The average of some irreducible character degrees (Q6636518)

Let \(G\) be a finite group of odd order, \(k\) a field that contains the primitive \(p\)-th roots of unity satisfying \(\mathbb{Q}_p \subseteq k \subseteq \mathbb{C}\). Depending on the value of \(p\) modulo \(3\), the authors provide upper bounds of the average of the character degrees and the average degrees of the characters whose values reside in \(k\), satisfaction of which guarantees the nilpotency of \(G\) (Theorem \(1.1\)).