A sufficient conditon for solvability of finite groups

Abstract

The following theorem is proved: Let $G$ be a finite group and ${\pi }_e(G)$ be the set of element orders in $G$. If ${\pi }_e(G)\cap \{2\}=\varnothing$; or ${\pi }_e(G)\cap \{3,4\}=\varnothing$; or ${\pi }_e(G)\cap \{3,5\}=\varnothing$, then $G$ is solvable. Moreover, using the intersection with ${\pi }_e(G)$ being empty set to judge $G$ is solvable or not, only the above three cases.

arXiv comment5 pages