On the other hand, "n""p" ≡ 1 (mod "p"). Hence, ("p" − 2)! ≡ 1 (mod "p"). So, ("p" − 1)! ≡ −1 (mod "p"). Frattini's argument shows that a Sylow subgroup of a normal subgroup provides a factorization of a finite group. A slight generalization known as Burnside's fusion theorem states that if "G" is a finite group with Sylow "p"-subgroup "P" and two subsets "A" and "B" normalized by "P", then "A" and "B" are "G"-conjugate if and only if they are "NG"("P")-conjugate. The proof is a simple application of Sylow's theorem: If "B"="Ag", then the normalizer of "B" contains not only "P" but also "Pg" (since "Pg" is contained in the normalizer of "Ag"). By Sylow's theorem "P" and "Pg" are conjugate not only in "G", but in the normalizer of "B". Hence "gh"−1 normalizes "P" for some "h" that normalizes "B", and then "A""gh"−1 = "B"h−1 = "B", so that "A" and "B" are "NG"("P")-conjugate. Burnside's fusion theorem can be used to give a more powerful factorization called a semidirect product: if "G" is a finite group whose Sylow "p"-subgroup "P" is contained in the center of its normalizer, then "G" has a normal subgroup "K" of order coprime to "P", "G" = "PK" and "P"∩"K" = {1}, that is, "G" is "p"-nilpotent. Less trivial applications of the Sylow theorems include the focal subgroup theorem, which studies the control a Sylow "p"-subgroup of the derived subgroup has on the structure of the entire group. This control is exploited at several stages of the classification of finite simple groups, and for instance defines the case divisions used in the Alperin–Brauer–Gorenstein theorem classifying finite simple groups whose Sylow 2-subgroup is a quasi-dihedral group.