mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, one can define a product of group subsets in a natural way. If ''S'' and ''T'' are

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

s of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

''G'', then their product is the subset of ''G'' defined by :

ST = \.

The subsets ''S'' and ''T'' need not be

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s for this product to be well defined. The

associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

of this product follows from that of the group product. The product of group subsets therefore defines a natural

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...

structure on the

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...

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 ''G'' if and only if ''ST'' = ''TS''.

Product of subgroups

If ''S'' and ''T'' are subgroups of ''G'', their product need not be a subgroup (for example, two distinct subgroups of order 2 in the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

on 3 symbols). 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. ( Walter Ledermann has called this fact the ''Product Theorem'', but this name, just like "Frobenius product" is by no means standard.) 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 Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

then the condition ''ST'' = ''TS'' is satisfied and the product is a subgroup.Nicholson, 2012, Theorem 5, p. 125 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'': :

, ST,  = \frac

Note that this applies even if neither ''S'' nor ''T'' is normal.

Modular law

The following modular law (for groups) holds for any ''Q'' a subgroup of ''S'', where ''T'' is any other arbitrary subgroup (and both ''S'' and ''T'' are subgroups of some group ''G''): :''Q''(''S'' ∩ ''T'') = ''S'' ∩ (''QT''). The two products that appear in this equality are not necessarily subgroups. If ''QT'' is a subgroup (equivalently, as noted above, if ''Q'' and ''T'' permute) then ''QT'' = ⟨''Q'' ∪ ''T''⟩ = ''Q'' ∨ ''T''; i.e., ''QT'' is the

join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...

of ''Q'' and ''T'' in the

lattice of subgroups In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their uni ...

of ''G'', and the modular law for such a pair may also be written as ''Q'' ∨ (''S'' ∩ ''T'') = ''S'' ∩ (''Q ∨ T''), which is the equation that defines a

modular lattice In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self- dual condition, ;Modular law: implies where are arbitrary elements in the lattice, ≤ is the partial order, and & ...

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 __NOTOC__ In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group ''G'' is called an Iwasawa group when every subgroup of ''G'' is permutable in ''G'' . proved ...

. The subgroup lattice of an Iwasawa group is thus a modular lattice, so these groups are sometimes called ''modular groups'' (although this latter term may have other meanings.) The assumption in the modular law for groups (as formulated above) 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'' ∩ (''QT'') = (''S'' ∩ ''Q'')(''S'' ∩ ''T'') is ''false''.

Product of subgroups with trivial intersection

In particular, if ''S'' and ''T'' intersect only in the identity, then every element of ''ST'' 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 In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed fro ...

. Even further, if ''S'' or ''T'' is normal in ''ST'', then ''ST'' coincides with the

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...

of ''S'' and ''T''. Finally, if both ''S'' and ''T'' are normal in ''ST'', then ''ST'' coincides with the

direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...

of ''S'' and ''T''. If ''S'' and ''T'' are subgroups whose intersection is the trivial subgroup (identity element) and additionally ''ST'' = ''G'', then ''S'' is called a

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...

of ''T'' and vice versa. By a (locally unambiguous)

abuse of terminology In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors a ...

, two subgroups that intersect only on the (otherwise obligatory) 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 subgroup ''N'' 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" (map to 1) all elements of ''K'' that happen to be in ''N''. Thus the correct answer is that ''NK''/''N'' is isomorphic with ''K''/(''N''∩''K''). This fact is sometimes called the

second isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exis ...

, (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, and has also been called the ''parallelogram rule'' by

Paul Moritz Cohn Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commutat ...

, who thus emphasized analogy with the

parallelogram rule In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of th ...

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 In group theory, a branch of mathematics, Frattini's argument is an important lemma in the structure theory of finite groups. It is named after Giovanni Frattini, who used it in a paper from 1885 when defining the Frattini subgroup of a group. T ...

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'' = ''N''_''G''(''P'')''N'', where ''N''_''G''(''P'') denotes the

normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...

of ''P'' in ''G''. (Note that the normalizer of ''P'' includes ''P'', so the intersection between ''N'' and ''N''_''G''(''P'') is at least ''P''.)

Generalization to semigroups

In a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

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 In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...

with addition as union (of subsets) and multiplication as product of subsets.

References