Suppose $G$ is a non-abelian finite group and $\pi(G)=\pi_1(G)\cup \pi_2(G)$ is a disjoint union of all primes dividing $|G|$. Suppose further that for every two elements $g_1,g_2\in G\setminus Z(G)$ if the order of $g_1\in \pi_1(G)$ and $g_2\in \pi_2(G)$, then $g_1g_2\neq g_2g_1$. Is there any result or idea about the structure of $G$?
P.S. If $Z(G)=1$, then $G$ has disconnected prime graph and its structure has recognized in J.S.Williams, Prime graph components of finite groups, Journal of Algebra 69 (1981), 487-513.
