View source for Complement (group theory)

In [[Mathematics|mathematics]], especially in the area of [[Algebra|algebra]] known as [[Group theory|group theory]], a '''complement''' of a [[Subgroup|subgroup]] ''H'' in a [[Group (mathematics)|group]] ''G'' is a subgroup ''K'' of ''G'' such that 
:<math>G = HK = \{ hk : h\in H, k\in K\} \text{ and } H\cap K = \{e\}.</math>
Equivalently, every element of ''G'' has a unique expression as a product ''hk'' where ''h'' ∈ ''H'' and ''k'' ∈ ''K''. This relation is symmetrical: if ''K'' is a complement of ''H'', then ''H'' is a complement of ''K''. Neither ''H'' nor ''K'' need be a [[Normal subgroup|normal subgroup]] of ''G''.

==Properties==
* Complements need not exist, and if they do they need not be unique. That is, ''H'' could have two distinct complements ''K''<sub>1</sub> and ''K''<sub>2</sub> in ''G''.
* If there are several complements of a normal subgroup, then they are necessarily [[Group isomorphism|isomorphic]] to each other and to the [[Quotient group|quotient group]].
* If ''K'' is a complement of ''H'' in ''G'' then ''K'' forms both a left and right transversal of ''H''. That is, the elements of ''K'' form a complete set of representatives of both the left and right [[Coset|coset]]s of ''H''.
* The [[Schur–Zassenhaus theorem]] guarantees the existence of complements of normal [[Hall subgroup]]s of [[Finite group|finite group]]s.

==Relation to other products==
Complements generalize both the [[Direct product of groups|direct product]] (where the subgroups ''H'' and ''K'' are normal in ''G''), and the [[Semidirect product|semidirect product]] (where one of ''H'' or ''K'' is normal in ''G'').  The product corresponding to a general complement is called the [[Zappa–Szép product#Internal Zappa–Szép products|internal Zappa–Szép product]].  When ''H'' and ''K'' are [[Trivial group|nontrivial]], complement subgroups factor a group into smaller pieces.

==Existence==
As previously mentioned, complements need not exist.

A '''''p''-complement''' is a complement to a [[Sylow subgroup|Sylow ''p''-subgroup]].  Theorems of [[Biography:Ferdinand Georg Frobenius|Frobenius]] and [[Biography:John G. Thompson|Thompson]] describe when a group has a [[Normal p-complement|normal ''p''-complement]].  [[Biography:Philip Hall|Philip Hall]] characterized finite [[Solvable group|soluble]] groups amongst finite groups as those with ''p''-complements for every [[Prime number|prime]] ''p''; these ''p''-complements are used to form what is called a Sylow system.

A '''Frobenius complement''' is a special type of complement in a [[Frobenius group]].

A [[Complemented group|complemented group]] is one where every subgroup has a complement.

==See also==
* [[Product of group subsets]]

==References==
*{{cite book | author=David S. Dummit & Richard M. Foote | title=Abstract Algebra | publisher=Wiley | year=2003 | isbn=978-0-471-43334-7}}
*{{cite book | author=I. Martin Isaacs | title=Finite Group Theory  | publisher=American Mathematical Society | year=2008 | isbn=978-0-8218-4344-4 }}

[[Category:Group theory]]



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