Graphs, partitions and classes of groups

摘要

If ${mathfrak X}$ is a class of groups, Delizia et al. (Bull Austral Math Soc 75:313–320, 2007) call a group G ${mathfrak X}$ -transitive (or an ${mathfrak XT}$ -group) if whenever ${langle a,brangle}$ and ${langle b,crangle}$ are in ${mathfrak X} langle a,crangle$ is also in ${mathfrak X}$ ( ${a,b,cin G}$ ). The structure of ${mathfrak XT}$ -groups has been investigated for a number of classes of groups, by Delizia, Moravec and Nicotera and others. A graph can be associated with a group in many ways. Delizia, Moravec and Nicotera introduce a graph which is a generalisation of the commuting graph of a group, but do not make use of the graph. We will use the properties of the graph to investigate further classes of groups and to obtain more detailed structural information.