Graphs, partitions and classes of groups |
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Authors: | Adolfo Ballester-Bolinches John Cossey |
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Institution: | 1. Departament d’àlgebra, Universitat de València, Dr. Moliner, 50, 46100, Burjassot, València, Spain 2. Department of Mathematics, Mathematical Sciences Institute, Australian National University, Canberra, ACT, 0200, Australia
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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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