Template:Natural number property implication
Suppose where the are pairwise distinct prime numbers. In other words, is a square-free number.
Then, is a solvability-forcing number: any Finite group (?) of order is a Solvable group (?), i.e., a Finite solvable group (?).
- Every Sylow subgroup is cyclic implies metacyclic
- Metacyclic implies solvable
The proof follows from facts (1) and (2), and the observation that in a group of square-free order, every nontrivial Sylow subgroup has prime order, and is hence cyclic.