Graphs, partitions and classes of groups

If X is a class of groups, Delizia et al. (Bull Austral Math Soc 75:313-320, 2007) call a group G X -transitive (or an XT -group) if whenever 〈 a, b 〉 and 〈 b, c〉 are in X 〈a, c〉 is also in X (a, b, c ∈ G). The structure of 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.