 This means "P" is a Sylow "p"-subgroup, which is abelian, as all diagonal matrices commute, and because Theorem 2 states that all Sylow "p"-subgroups are conjugate to each other, the Sylow "p"-subgroups of "GL"2("F""q") are all abelian.Since Sylow's theorem ensures the existence of p-subgroups of a finite group, it's worthwhile to study groups of prime power order more closely.Most of the examples use Sylow's theorem to prove that a group of a particular order is not simple.