For |G|=p^a m (p∤m): (I) Sylow p-subgroups of order p^a exist; (II) they are all conjugate; (III) their number n_p ≡ 1 (mod p) and divides m. Key tool for structural classification of finite groups.
For |G|=p^a m (p∤m): (I) Sylow p-subgroups of order p^a exist; (II) they are all conjugate; (III) their number n_p ≡ 1 (mod p) and divides m. Key tool for structural classification of finite groups.