between authors); it has also been called the diamond theorem by I. Martin Isaacs because of the shape of subgroup lattice involved, and has also been called the parallelogram rule by Paul Moritz Cohn, who thus emphasized analogy with the parallelogram rule for vectors because in the resulting subgroup lattice the two sides assumed to represent the quotient groups (SN) / N and S / (S ∩ N) are "equal" in the sense of isomorphism. Frattini's argument guarantees the existence of a product of subgroups (giving rise to the whole group) in a case where the intersection is not necessarily trivial (and for this latter reason the two subgroups are not complements). More specifically, if G is a finite group with normal subgroup N, and if P is a Sylow p-subgroup of N, then G = NG(P)N, where NG(P) denotes the normalizer of P in G. (Note that the normalizer of P includes P, so the intersection between N and NG(P) is at least P.) Generalization to semigroups In a semigroup S, the product of two subsets defines a structure of a semigroup on P(S), the power set of the semigroup S; furthermore P(S) is a semiring with addition as union (of subsets) and multiplication as product of subsets. See also Central product Double coset References Group products Operations on structures