On the \(\Pi\)-property of subgroups of finite groups. (Q500850)

Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is said to satisfy the \(\Pi\)-property in \(G\) if every prime dividing \(|G:N_G(HK\cap L)|\) also divides the order of \((HK\cap L)/K\), for every chief factor \(L/K\) of \(G\). In this note the authors prove that the finite group \(G\) is soluble if and only if all maximal subgroups of \(G\) satisfy the \(\Pi\)-property in \(G\). This answers affirmatively the Question 5.2 posed by \textit{B. Li} [in J. Algebra 334, No. 1, 321-337 (2011; Zbl 1248.20020)].