Mathematics > Group Theory
[Submitted on 19 Sep 2022 (v1), last revised 2 Oct 2022 (this version, v2)]
Title:The average character degree of finite groups and Gluck's conjecture
View PDFAbstract:We prove that the order of a finite group $G$ with trivial solvable radical is bounded above in terms of ${\rm acd}(G)$, the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of ${\rm acd}(G)$, but we show that in certain cases it is bounded in terms of the degrees of the irreducible characters of $G$ that lie over a linear character of the Fitting subgroup. This leads us to propose a refined version of Gluck's conjecture.
Submission history
From: Alexander Moretó [view email][v1] Mon, 19 Sep 2022 18:05:39 UTC (10 KB)
[v2] Sun, 2 Oct 2022 11:25:24 UTC (10 KB)
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