Editing Product of group subsets

In [[mathematics]], one can define a '''product of group subsets''' in a natural way. If ''S'' and ''T'' are [[subset]]s of a [[group (mathematics)|group]] ''G'', then their product is the subset of ''G'' defined by
:<math>ST = \{st : s \in S \text{ and } t\in T\}.</math>
The subsets ''S'' and ''T'' need not be [[subgroup]]s 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 ''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]] on 3 symbols). This product is sometimes called the ''Frobenius product''.<ref name="Ballester-BolinchesEsteban-Romero2010p1">{{cite book|author1=Adolfo Ballester-Bolinches|author2=Ramon Esteban-Romero|author3=Mohamed Asaad|title=Products of Finite Groups|url=https://archive.org/details/productsfinitegr00masa|url-access=limited|year=2010|publisher=Walter de Gruyter|isbn=978-3-11-022061-2|page=[https://archive.org/details/productsfinitegr00masa/page/n13 1]}}</ref> In general, the product of two subgroups ''S'' and ''T'' is a subgroup if and only if ''ST'' = ''TS'',<ref name="Nicholson2012">{{cite book|author=W. Keith Nicholson|title=Introduction to Abstract Algebra|year=2012|publisher=John Wiley & Sons|edition=4th|isbn=978-1-118-13535-8|at=Lemma 2, p. 125}}</ref> and the two subgroups are said to [[permutable subgroup|permute]]. ([[Walter Ledermann]] has called this fact the ''Product Theorem'',<ref>Walter Ledermann, ''Introduction to Group Theory'', 1976, Longman, {{ISBN|0-582-44180-3}}, p. 52</ref> but this name, just like "Frobenius product" is by no means standard.) In this case, ''ST'' is the group [[generating set of a group|generated]] by ''S'' and ''T''; i.e., ''ST'' = ''TS'' = ⟨''S'' ∪ ''T''⟩.

If either ''S'' or ''T'' is [[normal subgroup|normal]] then the condition ''ST'' = ''TS'' is satisfied and the product is a subgroup.<ref name="NT5">Nicholson, 2012, Theorem 5, p. 125</ref><ref name="Wallace1998">{{cite book|author=David A.R. Wallace|title=Groups, Rings and Fields|year=1998|publisher=Springer Science & Business Media|isbn=978-3-540-76177-8|at=Theorem 14, p. 123}}</ref> If both ''S'' and ''T'' are normal, then the product is normal as well.<ref name="NT5"/>

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'':
:<math>|ST| = \frac{|S||T|}{|S\cap T|}</math>
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 (mathematics)|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'' ∩ ''T'') = ''S'' ∩ (''Q ∨ T''), 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''<ref>Ballester-Bolinches, Esteban-Romero, Asaad, p. 24</ref> (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''.<ref name="Robinson1996">{{cite book|author=Derek Robinson|title=A Course in the Theory of Groups|year=1996|publisher=Springer Science & Business Media|isbn=978-0-387-94461-6|page=15}}</ref><ref name="Cohn2000p248">{{cite book|author=Paul Moritz Cohn|title=Classic algebra|url=https://archive.org/details/classicalgebra00cohn|url-access=limited|year=2000|publisher=Wiley|isbn=978-0-471-87731-8|pages=[https://archive.org/details/classicalgebra00cohn/page/n259 248]}}</ref>

=== 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]].  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 groups|direct product]] 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 (group theory)|complement]] of ''T'' and vice versa.

By a (locally unambiguous) [[abuse of terminology]], two subgroups that intersect only on the (otherwise obligatory) identity are sometimes called [[Disjoint sets|disjoint]].<ref>{{cite book|author=L. Fuchs|title=Infinite Abelian Groups. Volume I|year=1970|publisher=Academic Press|isbn=978-0-08-087348-0|page=37}}</ref>

=== 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]],<ref name="Saracino1980">{{cite book|author=Dan Saracino|title=Abstract Algebra: A First Course|url=https://archive.org/details/abstractalgebraf00sara_691|url-access=limited|year=1980|publisher=Addison-Wesley|isbn=0-201-07391-9|page=[https://archive.org/details/abstractalgebraf00sara_691/page/n127 123]}}</ref> (although the numbering of these theorems sees some variation between authors); it has also been called the ''diamond theorem'' by [[Martin Isaacs|I. Martin Isaacs]] because of the shape of subgroup lattice involved,<ref name="Isaacs1994">{{cite book|author=I. Martin Isaacs|title=Algebra: A Graduate Course|url=https://archive.org/details/algebragraduatec00isaa|url-access=limited|year=1994|publisher=American Mathematical Soc.|isbn=978-0-8218-4799-2|page=[https://archive.org/details/algebragraduatec00isaa/page/n45 33]}}</ref> 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'')&nbsp;/&nbsp;''N'' and ''S''&nbsp;/&nbsp;(''S''&nbsp;∩&nbsp;''N'') are "equal" in the sense of isomorphism.<ref name="Cohn2000p245">{{cite book|author=Paul Moritz Cohn|author-link=Paul Moritz Cohn|title=Classic Algebra|url=https://archive.org/details/classicalgebra00cohn|url-access=limited|year=2000|publisher=Wiley|isbn=978-0-471-87731-8|page=[https://archive.org/details/classicalgebra00cohn/page/n256 245]}}</ref>

[[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|Sylow ''p''-subgroup]] of ''N'', then ''G'' = ''N''<sub>''G''</sub>(''P'')''N'', where ''N''<sub>''G''</sub>(''P'') denotes the [[centralizer and normalizer|normalizer]] of ''P'' in ''G''. (Note that the normalizer of ''P'' includes ''P'', so the intersection between ''N'' and ''N''<sub>''G''</sub>(''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.<ref name="Pin1989">{{cite book|author=Jean E. Pin|title=Formal Properties of Finite Automata and Applications: LITP Spring School on Theoretical Computer Science, Ramatuelle, France, May 23–27, 1988. Proceedings|year=1989|publisher=Springer Science & Business Media|isbn=978-3-540-51631-6|pages=35}}</ref>

==See also==
* [[Central product]]
* [[Double coset]]

==References==
{{Reflist}}
*{{cite book
 | first      = Joseph
 | last       = Rotman
 | year       = 1995
 | title      = An Introduction to the Theory of Groups
 | edition    = 4th
 | publisher  = Springer-Verlag
 | isbn       = 0-387-94285-8
}}

[[Category:Group products]]
[[Category:Operations on structures]]