Some properties of finite groups with some (semi-\(p\)-)cover-avoiding subgroups. (Q1004471)

The authors introduce, for a prime \(p\), the notion of semi-\(p\)-cover-avoiding subgroup: this is a subgroup \(H\) of a finite group \(G\) with the property that there exists a chief series of \(G\) such that \(H\) either covers or avoids each chief factor whose order is divisible by the prime \(p\). They use this notion in order to obtain sufficient conditions for a group to be \(p\)-nilpotent or supersolvable. A typical example is Theorem 3.1: ``Let \(G\) be a finite group and let \(p\in\pi(G)\) such that \((|G|,p^2-1)=1\). If \(N\) is a normal subgroup of \(G\) such that \(G/N\) is \(p\)-nilpotent and every cyclic subgroup of \(N\) with order at most \(p^k\) (here \(k=1+\tfrac{1+(-1)^p}{2}\)) is semi-\(p\)-cover-avoiding in \(G\), then \(G\) is \(p\)-nilpotent.''