# Complement (group theory)

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In [mathematics](/source/mathematics), especially in the area of [algebra](/source/algebra) known as [group theory](/source/group_theory), a '''complement''' of a [subgroup](/source/subgroup) ''H'' in a [group](/source/group_(mathematics)) ''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](/source/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 [isomorphic](/source/group_isomorphism) to each other and to the [quotient group](/source/quotient_group).
* If ''K'' is a complement of ''H'' in ''G'' then ''K'' forms both a left and right [transversal](/source/transversal_(group_theory)) of ''H''. That is, the elements of ''K'' form a complete set of representatives of both the left and right [coset](/source/coset)s of ''H''.
* The [Schur–Zassenhaus theorem](/source/Schur%E2%80%93Zassenhaus_theorem) guarantees the existence of complements of normal [Hall subgroup](/source/Hall_subgroup)s of [finite group](/source/finite_group)s.

==Relation to other products==
Complements generalize both the [direct product](/source/direct_product_of_groups) (where the subgroups ''H'' and ''K'' are normal in ''G''), and the [semidirect product](/source/semidirect_product) (where one of ''H'' or ''K'' is normal in ''G'').  The product corresponding to a general complement is called the [internal Zappa–Szép product](/source/Zappa%E2%80%93Sz%C3%A9p_product).  When ''H'' and ''K'' are [nontrivial](/source/trivial_group), 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 ''p''-subgroup](/source/Sylow_subgroup).  Theorems of [Frobenius](/source/Ferdinand_Georg_Frobenius) and [Thompson](/source/John_G._Thompson) describe when a group has a [normal ''p''-complement](/source/normal_p-complement).  [Philip Hall](/source/Philip_Hall) characterized finite [soluble](/source/solvable_group) groups amongst finite groups as those with ''p''-complements for every [prime](/source/prime_number) ''p''; these ''p''-complements are used to form what is called a [Sylow system](/source/Sylow_system).

A '''Frobenius complement''' is a special type of complement in a [Frobenius group](/source/Frobenius_group).

A [complemented group](/source/complemented_group) is one where every subgroup has a complement.

==See also==
* [Product of group subsets](/source/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 |authorlink = Martin Isaacs| title=Finite Group Theory  | publisher=American Mathematical Society | year=2008 | isbn=978-0-8218-4344-4 }}

Category:Group theory

{{group-theory-stub}}

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Adapted from the Wikipedia article [Complement (group theory)](https://en.wikipedia.org/wiki/Complement_(group_theory)) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Complement_(group_theory)?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.