Finite groups with many \(p\)-regular conjugacy classes (Q6187325)

Let \(G\) be a finite group. The probability that two randomly chosen elements of \(G\) commute is given by \(d(G)=|\{(x,y) \in G^{2} \mid [x,y]=1\}|\cdot |G|^{-2}\) and an elementary argument shows that \(d(G)=k(G)\cdot|G|^{-1}\), where \(k(G)\) is the number of conjugacy classes of \(G\). \textit{G. W. Gustafson} [Am. Math. Mon. 80, 1031--1034 (1973; Zbl 0276.60013)] proved that if \(d(G)>5/8\), then \(G\) is abelian. For every set \(\pi\) of primes, let \(k_{\pi}(G)\) be the number of conjugacy classes of \(\pi\)-elements of \(G\). The local invariant \(d_{\pi}(G)=k_{\pi}(G) \cdot |G|_{\pi}^{-1}\) captures information about the \(\pi\)-structure of the group. Indeed, \textit{A. Maróti} and \textit{H. N. Nguyen} proved in [Arch. Math. 102, No. 2, 101--108 (2014; Zbl 1295.20016)] that if \(d_{\pi}(G) > 5/8\), then \(G\) contains an abelian Hall \(\pi\)-subgroup. If \(\pi=p'\), then \(k_{p'}(G)\) is the number of conjugacy classes of elements whose order is not divisible by \(p\), the so-called \(p\)-regular elements. It is also the number of inequivalent irreducible \(p\)-modular representations of \(G\). Viewed in this light, the invariant \(d_{p'}(G)\) holds special interest because it reflects the \(p\)-modular representation theory of \(G\). In the paper under review, the author studies the structure of finite groups with a large number of \(p\)-regular conjugacy classes or, equivalently, a large number of irreducible \(p\)-modular representations. In particular he proves that, if \(d_{p'}(G)>1/(p-1)\), then \(G\) is \(p\)-solvable. Moreover, \(G\) has \(p\)-length at most \(2\) and the number of nonabelian simple \(p\)-factors in a composition series for \(G\) is strictly less than \(\ln(p-1)/ \ln(12)\). In particular, if \(p\leq 13\), then \(G\) is solvable. Two other interesting results are: if \(d_{2'}(G)>4/5\), then \(G\) is solvable with \(2\)-length at most \(4\) (see Theorem 1.2) and if \(d_{p'}(G)> 2/(p-1)\), then \(G\) has a normal Sylow \(p\)-subgroup (see Theorem 1.3). Finally the author proves Theorem 1.4: If \(G\) is \(p\)-solvable of \(p\)-length \(k\), then \(d_{p'}(G) \leq \big (2/(p+1) \big )^{k-1}\).