Editing Sylow theorems (section)

=== Cyclic group orders ===
Some non-prime numbers ''n'' are such that every group of order ''n'' is cyclic.  One can show that ''n'' = 15 is such a number using the Sylow theorems:  Let ''G'' be a group of order 15 = 3 · 5 and ''n''<sub>3</sub> be the number of Sylow 3-subgroups. Then ''n''<sub>3</sub> <math>\mid</math> 5 and ''n''<sub>3</sub> ≡ 1 (mod 3). The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be [[normal subgroup|normal]] (since it has no distinct conjugates). Similarly, ''n''<sub>5</sub> must divide 3, and ''n''<sub>5</sub> must equal 1 (mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are [[coprime]], the intersection of these two subgroups is trivial, and so ''G'' must be the [[internal direct product]] of groups of order 3 and 5, that is the [[cyclic group]] of order 15. Thus, there is only one group of order 15 ([[up to]] isomorphism).