Abstract
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 \({\langle a,b\rangle}\) and \({\langle b,c\rangle}\) are in \({\mathfrak X} \langle a,c\rangle\) is also in \({\mathfrak X}\) (\({a,b,c\in G}\)). 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.
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Communicated by John S. Wilson.
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Ballester-Bolinches, A., Cossey, J. Graphs, partitions and classes of groups. Monatsh Math 166, 309–318 (2012). https://doi.org/10.1007/s00605-010-0263-3
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DOI: https://doi.org/10.1007/s00605-010-0263-3