A new lower bound for the number of conjugacy classes (Q6583750)

Let \(G\) be a finite group, \(k(G)\) the number of conjugacy classes of \(G\) and \(p \in \pi(G)\). If \(G\) is soluble, then \textit{L. Héthelyi} and \textit{B. Külshammer} [Bull. Lond. Math. Soc. 32, No. 6, 668--672 (2000; Zbl 1024.20016)] proved that \(k(G) \geq 2 \sqrt{p-1}\) and this bound is sharp.\N\NLet \(p > 3\) be a prime and let \(a, b \in \mathbb{N}\) be such that \(p-1=ab\) and such that \(|a-b|\) is minimal. Let \(K_{a}=C_{p} \rtimes C_{a}\) such that \(C_{K_{a}}(C_{p})=C_{p}\), and let \(K_{b}=C_{p} \rtimes C_{b}\) such that \(C_{K_{b}}(C_{p})=C_{p}\). In this paper, the authors prove that, assuming the McKay conjecture, for every \(G\) with \(p \in \pi(G)\), one has \(k(G) \geq a+b\) with equality if and only if \(G \simeq K_{a}\) or \(G \simeq K_{b}\).\N\NIf \(G\) is solvable, the authors note that the result above is unconditionally true (as is observed in [\textit{L. Héthelyi} and \textit{B. Külshammer}, J. Algebra 270, No. 2, 660--669 (2003; Zbl 1047.20016), Remark (ii)]).