Coprime implies one is solvable

Statement
If two finite groups have relatively prime orders, then one of the groups is solvable.

Applications

 * Hall retract implies order-conjugate (part of the Schur-Zassenhaus theorem)
 * Sylow's theorem with operators

For a complete list of applications, see Category:Facts about groups of coprime order whose proof requires the assumption that one of them is solvable.

Proof
This follows directly from the Feit-Thompson theorem.