# Finite group is solvable iff identity element is not expressible as product of three elements of pairwise coprime order

From Groupprops

## Definition

Let be a solvable group. Then, the following are equivalent for :

- is a solvable group (in particular, it is a finite solvable group).
- There do not exist three non-identity elements such that the orders of are pairwose relatively prime, and is the identity element.