Editing Product of group subsets (section)

=== 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''.)