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

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


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

Related facts

 * Solvability is 2-local for finite groups
 * Classification of finite minimal simple groups