In
mathematics, especially in the area of
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
known as
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, a complement of a
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 subgrou ...
''H'' in 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'' is a subgroup ''K'' of ''G'' such that
:
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 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''
1 and ''K''
2 in ''G''.
* If there are several complements of a normal subgroup, then they are necessarily
isomorphic to each other and to the
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
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of ''H''.
* The
Schur–Zassenhaus theorem
The Schur–Zassenhaus theorem is a theorem in group theory which states that if G is a finite group, and N is a normal subgroup whose order is coprime to the order of the quotient group G/N, then G is a semidirect product (or split extension) ...
guarantees the existence of complements of normal
Hall subgroup
In mathematics, specifically group theory, a Hall subgroup of a finite group ''G'' is a subgroup whose order is coprime to its index. They were introduced by the group theorist .
Definitions
A Hall divisor (also called a unitary divisor) of ...
s of
finite groups.
Relation to other products
Complements generalize both 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 ...
(where the subgroups ''H'' and ''K'' are normal in ''G''), and 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 ...
(where one of ''H'' or ''K'' is normal in ''G''). The product corresponding to a general complement is called the
internal Zappa–Szép product. When ''H'' and ''K'' are
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 ''p''-subgroup. Theorems of
Frobenius and
Thompson describe when a group has a
normal ''p''-complement.
Philip Hall characterized finite
soluble groups amongst finite groups as those with ''p''-complements for every
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''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 is one where every subgroup has a complement.
See also
*
Product of group subsets
References
*
*
Group theory
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