In mathematics, one can define a product of group subsets in a natural way. If S and T are subsets of a groupG, then their product is the subset of G defined by The subsets S and T need not be subgroups for this product to be well defined. The associativity of this product follows from that of the group product. 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. The product of two subgroups S and T of a group G is itself a subgroup of Gif and only ifST = TS.
Product of subgroups
If S and T are subgroups of G, their product need not be a subgroup. This product is sometimes called the Frobenius product. In general, the product of two subgroups S and T is a subgroup if and only if ST = TS, and the two subgroups are said to permute. In this case, ST is the group generated by S and T; i.e., ST = TS = ⟨S ∪ T⟩. If either S or T is normal then the condition ST = TS is satisfied and the product is a subgroup. If both S and T are normal, then the product is normal as well. 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 neitherS nor T is normal.
Modular law
The following modular law holds for any Q a subgroup of S, where T is any other arbitrary subgroup : The two products that appear in this equality are not necessarily subgroups. If QT is a subgroup then QT = ⟨Q ∪ T⟩ = Q ∨ T; i.e., QT is the join of Q and T in the lattice of subgroups of G, and the modular law for such a pair may also be written as Q ∨ = S ∩, which is the equation that defines a modular lattice if it holds for any three elements of the lattice with Q ≤ S. In particular, since normal subgroups permute with each other, they form a modular sublattice. A group in which every subgroup permutes is called an Iwasawa group. The subgroup lattice of an Iwasawa group is thus a modular lattice, so these groups are sometimes called modular groups The assumption in the modular law for groups that Q is a subgroup of S is essential. If Q is not a subgroup of S, then the tentative, more general distributive property that one may consider S ∩ = is false.
In particular, if S and T intersect only in the identity, then every element ofST has a unique expression as a product st with s in S and t in T. If S and T also commute, then ST is a group, and is called a Zappa–Szép product. Even further, if S or T is normal in ST, then ST coincides with the semidirect product of S and T. Finally, if both S and T are normal in ST, then ST coincides with the direct product of S and T. If S and T are subgroups whose intersection is the trivial subgroup and additionally ST = G, then S is called a complement of T and vice versa. By a abuse of terminology, two subgroups that intersect only on the identity are sometimes called disjoint.
Product of subgroups with non-trivial intersection
A question that arises in the case of a non-trivial intersection between a normal subgroupN and a subgroup K is what is the structure of the quotient NK/N. Although one might be tempted to just "cancel out" N and say the answer is K, that is not correct because a homomorphism with kernel N will also "collapse" all elements of K that happen to be in N. Thus the correct answer is that NK/N is isomorphic with K/. This fact is sometimes called the second isomorphism theorem, ; 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 / N and S / are "equal" in the sense of isomorphism. Frattini's argument guarantees the existence of a product of subgroups in a case where the intersection is not necessarily trivial. More specifically, if G is a finite group with normal subgroup N, and if P is a Sylow p-subgroup of N, then G = NGN, where NG denotes the normalizer of P in G.
Generalization to semigroups
In a semigroup S, the product of two subsets defines a structure of semigroup on P, the power set of the semigroup S; furthermore P is a semiring with addition as union and multiplication as product of subsets.
