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  1. ; ; ; ; ; ; (, Bulletin of the Australian Mathematical Society)
    null (Ed.)
    Abstract: Morgan and Parker proved that if G is a group with 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 Z(G)=1. We also prove that if G is solvable and G/Z(G) is either a Frobenius group or a 2-Frobenius group, then the commuting graph of G is disconnected. 
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