Extending Morgan and Parker's results about commuting graphs

TL;DR

This paper extends previous results on the structure of commuting graphs in groups by relaxing conditions on the center and derived subgroup, and characterizes the connectivity and diameter of these graphs under new assumptions.

Contribution

It generalizes Morgan and Parker's results by replacing the condition Z(G)=1 with G' ∩ Z(G)=1 and explores the connectivity of commuting graphs in solvable groups with specific quotient structures.

Findings

Replacing Z(G)=1 with G' ∩ Z(G)=1 broadens the applicability of the results.

Connected commuting graphs of certain solvable groups have diameter at most 8.

Disconnection of commuting graphs occurs in solvable groups with specific quotient structures.

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.