Extending Morgan and Parker's results about commuting graphs

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 $G^{\prime }\cap Z(G)=1$. 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.

10 pages, 4 figures