The product of group subsets therefore defines a natural monoid structure on the power set of G. A lot more can be said in the case where S and T are subgroups.
[4] If S and T are finite subgroups of a group G, then ST is a subset of G of size |ST| given by the product formula: Note that this applies even if neither S nor T is normal.
By a (locally unambiguous) abuse of terminology, two subgroups that intersect only on the (otherwise obligatory) identity are sometimes called disjoint.
This fact is sometimes called the second isomorphism theorem,[10] (although the numbering of these theorems sees some variation between authors); it has also been called the diamond theorem by I. Martin Isaacs because of the shape of subgroup lattice involved,[11] 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.
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.) 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.