A criterion for \(\pi'\)-closures of \(\pi\)-homogeneous groups (Q2748805)

Let \(G\) be a finite group, \(\pi(G)\) the set of prime divisors of \(|G|\) and \(\pi\subseteq\pi(G)\). A positive integer \(n\) is called \(\pi\)-number if the prime factors of \(n\) are all contained in \(\pi\). \(G\) is called \(\pi\)-homogeneous group if \(N_G(H)/C_G(H)\) is a \(\pi\)-subgroup for every \(\pi\)-subgroup \(H\) of \(G\), and \(G\) is called \(\pi'\)-closed group if \(G\) has a normal Hall \(\pi'\)-subgroup. It is well known that \(\pi'\)-closed groups are \(\pi\)-homogeneous. The converse, in general, does not hold. Using the classification of the minimal simple groups, the author shows that if \(2^p-1\) is not a \(\pi\)-number for every prime \(p\), then \(G\) is \(\pi'\)-closed if and only if \(G\) is \(\pi\)-homogeneous.