On the average character degree and the average class size in finite groups. (Q479801)

Let \(G\) be a finite group, and let \(\mathrm{acd}(G)\) denote the average of the degrees of the irreducible characters of \(G\). The paper is another contribution to the (still very young) study of what \(\mathrm{acd}(G)\) reveals about the structure of the group. So far most results on this topic are of the flavor that if \(\mathrm{acd}(G)\) is small, then the structure of the group is restricted. For example, if \(\mathrm{acd}(G)\leq\frac{3}{2}\), then \(G\) is supersolvable. The paper continues in this vain as follows. Given a prime \(r\), determines the strongest possible lower bound for \(\mathrm{acd}(G)\) among all non-\(r\)-solvable finite groups. This bound turns out to be quadratic in \(r\), and equality occurs if and only if \(G\) is a direct product of an abelian group and \(\mathrm{PSL}(2,q)\), where \(q=5\) if \(r=2,3\), and \(q=r\) if \(r\geq 5\). As a consequence, if \(G\) is nonsolvable, then \(\mathrm{acd}(G)\geq\frac{16}{5}\), a result that had already been obtained by \textit{A. Moretó} and \textit{H. N. Nguyen} [Bull. Lond. Math. Soc. 46, No. 3, 454-462 (2014; Zbl 1335.20007)]. The author also considers the corresponding questions for the average conjugacy class size in place of the average character degree and also obtains quadratic lower bounds for these. As a consequence, if the average class size is \(\leq\frac{40}{3}\), then it is either 12 or \(\frac{40}{3}\), and the structure of the group in this case is described. The paper contains a number of other interesting results related to the subject; the proofs depend on CFSG.