Square-free implies solvability-forcing
From Groupprops
Template:Natural number property implication
Statement
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 (?).
Facts used
Proof
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.
References
Textbook references
- The Theory of Groups by Marshall Hall, Jr., Page 148, Corollary 9.4.1, ^{More info}