Commuting Graphs of Finite Groups

The commuting graph of a finite group is defined to have the nontrivial elements of the group as its vertices, and an edge joining each commuting pair of elements. We explore the structure of the commuting graph for a variety of groups. In particular, the diameter of the commuting graph of the symmetric group S n is precisely described, based on the nature of n and n − 1. Furthermore the connected components of this graph are completely classified. We continue by establishing upper bounds on the diameter of the commuting graph for a certain class of solvable groups. Finally, we provide a structure theorem for groups of order p a q b that consist strictly of pand q-elements, including a description of the commuting graph of such a group.

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