# A sufficient conditon for solvability of finite groups

## TL;DR

This paper proves that finite groups lacking elements of certain orders (2, 3, 4, 5) are necessarily solvable, establishing specific conditions based on element order sets.

## Contribution

The paper introduces new sufficient conditions for the solvability of finite groups based on the absence of elements with specific orders.

## Key findings

- If no elements of order 2 exist, the group is solvable.
- If no elements of orders 3 or 4 exist, the group is solvable.
- If no elements of orders 3 or 5 exist, the group is solvable.

## 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 $\pi_e(G) \cap \{2\}=\emptyset$; or $\pi_e(G) \cap \{3, 4\}=\emptyset$; or $\pi_e(G) \cap \{3,5\}=\emptyset$, 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.