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- W3137569063 abstract "Morgan and Parker have proved that if $G$ is a group satisfying the condition that $Z(G) = 1$, then the connected components of the commuting graph of $G$ have diameter at most $10$. Parker has 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." @default.
- W3137569063 created "2021-03-29" @default.
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- W3137569063 date "2021-03-12" @default.
- W3137569063 modified "2023-09-27" @default.
- W3137569063 title "Extending Morgan and Parker's results about commuting graphs" @default.
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