On the \(NC\)-groups (Q1179838)

Let \(G\) be a finite group and let \(\pi\subseteq \pi(G)\) be a set of primes. \(G\) is called an \(NC_ \pi\)-group if for every \(p\in \pi\) and \(P\in \hbox{Syl}_ p(G)\) one has \(N_ G(Z(P))=C_ G(Z(P))\). If \(\pi=\pi(G)\), one obtains the notion of \(NC\)-group. There is no hope to determine all \(NC\)-groups, because P. Hauck showed [unpublished communication to the authors] that every finite group can be embedded in an \(NC\)-group. The authors restrict themselves to the case of solvable \(NC\)-groups and show that in certain very special conditions --- given mainly in terms of supersolvability of certain factors of the Fitting series --- the center of a solvable \(NC\)-group is nontrivial.