Extensions with overlap.
An old result of \textit{P. Hall} states that if a group, \(G\), has a normal subgroup, \(N\), which is nilpotent and so that the quotient group \(G/N'\) is also nilpotent, then \(G\) is nilpotent, [Ill. J. Math. 2, 787-801 (1959; Zbl 0084.25602)]. It is not true that if \(G\) and \(G/N\) are nilpotent then \(G\) is. The main part of the paper is concerned with classes of groups defined by some permutability condition on classes of subgroups. They show that in these cases there is an equivalent to the derived group which plays the same role as in Hall's Theorem. A subgroup \(H\) of a group \(G\) is called S-permutable if \(HP=PH\) for all Sylow subgroups \(P\) of \(G\). Further PST-groups are those in which being an S-group is a transitive relation. For a given word, say \(w\), we define \(w(G)\) to be the subgroup of \(G\) generated by all the elements of \(G\) which can be obtained by substituting elements of \(G\) in \(w\). So, for example, \(G'\) is \(w(G)\) for the word \(x^{-1}y^{-1}xy\). Theorem A says that if \(G\) has a normal subgroup \(N\) and both \(N\) and \(G/w(N)\) are soluble PST-groups then so is \(G\), where \(w=[[x,y],[z,t]]\). There are number of results which relate groups with transitivity properties to conditions on the Sylow subgroups. In particular, a finite soluble group is a PST-group if and only if for every prime \(p\) any \(p\)-subgroup \(K\) of \(G\) is S-permutable in \(N_G(K)\), [\textit{A. Ballester-Bolinches, R. Esteban-Romero}, J. Algebra 251, No. 2, 727-738 (2002; Zbl 1010.20013)]. This enables the authors to use local conditions and in Theorem B they find appropriate words for a number of classes which are defined in terms of these local properties. They finish the article with the following curious result which does not depend on the finiteness of the group. Theorem E: Let \(G\) be a group and \(N\) a normal subgroup of \(G\). Assume that there exist integers \(k\) and \(m\) so that \([G_k,N]N'=[G_{k+1},N]N'\) and \([G_m,N']\subseteq N_4\). Then there is an integer \(t\) so that \([G_t,N']=1\).
scientific article; zbMATH DE number 5696952
scientific article