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.

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摩根和帕克关于通勤图的扩展结果

摩根和帕克证明，如果G是一个组${\textbf{Z}(G)} = 1$，则通勤图的连通分量为G最多有直径10 美元. 帕克证明，如果，此外，G是可解的，则通勤图为G当且仅当断开连接G是 Frobenius 群或$2$-Frobenius 群，如果通勤图G是连通的，那么它的直径最多为$8$. 我们证明假设$Z (G) = 1$在这些结果中可以替换为$G' \cap {\textbf{Z}(G)} = 1$. 我们还证明如果G是可解的并且$G/{\textbf{Z}(G)}$是 Frobenius 群或$2$-Frobenius 群，然后是的通勤图G已断开连接。