Geodesic
A characterization via graphs of the soluble groups in which permutability is transitive
Algebra and discrete mathematics, no. 4 (2009), pp. 10-17.

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There are different ways to associate to a group a certain graph. In this context, it is interesting to ask for the relations between the structure of the group, given in group-theoretical terms, and the structure of the graphs, given in the language of graph theory. In this paper we recall some properties of the groups in which permutability is a transitive relation and present a new characterisation of the class of soluble groups in which permutability is a transitive relation in graph-theoretical terms.
Keywords: Finite soluble group; PT-group; permutable subgroup; complete graph. 
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A. Ballester-Bolinches; John Cossey; R. Esteban-Romero. A characterization via graphs of the soluble groups in which permutability is transitive. Algebra and discrete mathematics, no. 4 (2009), pp. 10-17. http://geodesic.mathdoc.fr/item/ADM_2009_4_a2/