Graphs, partitions and classes of groups.
If \(\mathcal X\) is a class of groups, a group \(G\) is called \(\mathcal X\)-transitive (or an \(\mathcal XT\)-group) if, for \(a,b,c\in G\), whenever \(\langle a,b\rangle\) and \(\langle b,c\rangle\) are in \(\mathcal X\), \(\langle a,c\rangle\) is also in \(\mathcal X\). To a group \(G\) it can be associated a graph \(\Gamma_{\mathcal X}(G)\) whose vertices are the non-trivial elements of \(G\) and in which \(a\) and \(b\) are adjacent if and only if \(\langle a,b\rangle\in\mathcal X\). The authors notice that, as a consequence of the classification of groups with a partition, if \(G\) is an \(\mathcal XT\)-group, then the connected components of \(\Gamma_{\mathcal X}(G)\) form a normal partition of \(G\setminus\{1\}\) and either \(\Gamma_{\mathcal X}(G)\) is connected or \(G\) is one of the following groups: a Frobenius group, \(\text{PSL}(2,2^h)\), \(Sz(2^h)\). The aim of the authors is to describe more completely the groups that occur as \(\mathcal XT\)-group for a number of classes \(\mathcal X\). Their main result is the following: let \(\mathcal X\) be a 2-recognizable class of soluble groups containing the class of the Abelian groups and the class \(\mathcal D\) of groups with cyclic Sylow subgroups; then the class of all \(\mathcal XT\)-groups is contained in \(\mathcal X\cup\mathcal F_{\mathcal X}\), where \(\mathcal F_{\mathcal X}\) is the class of Frobenius groups with the property that the kernel \(K\) is an \(\mathcal X\)-group, \(G/K\in\mathcal D\) and for each prime \(p\mid|G/K|\) and each prime \(q\mid|K|\), \(p\) does not divide \(q-1\).
scientific article; zbMATH DE number 6050538
scientific article