EXTENDING RESULTS OF MORGAN AND PARKER ABOUT COMMUTING GRAPHS
Abstract Morgan and Parker proved that if G is a group with ${\textbf{Z}(G)} = 1$ , then the connected components of the commuting graph of G have diameter at most $10$ . Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a $2$ -Frobenius group, and if the commuting graph of G is connected, then its diameter is at most $8$ . We prove that the hypothesis $Z (G) = 1$ in these results can be replaced with $G' \cap {\textbf{Z}(G)} = 1$ . We also prove that if G is solvable and $G/{\textbf{Z}(G)}$ is either a Frobenius group or a $2$ -Frobenius group, then the commuting graph of G is disconnected.
2006 ◽
Vol 17
(03)
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pp. 677-701
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2015 ◽
Vol 27
(10)
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pp. 2658-2671
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1998 ◽
Vol 20
(6)
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pp. 619-636
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