In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in the area of
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
known as
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, a complement of a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
''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 iden ...
''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
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
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
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to each other and to the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
.
* 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 guarantees the existence of complements of normal
Hall subgroups of
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s.
Relation to other products
Complements generalize both the
direct product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
(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. It is usually denoted with the symbol . There are two closely related concepts of semidirect product:
* an ''inner'' sem ...
(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
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group (mathematics), group, topological space). The n ...
, 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 Frobenius is a surname. Notable people with the surname include:
* Ferdinand Georg Frobenius (1849–1917), mathematician
** Frobenius algebra
** Frobenius endomorphism
** Frobenius inner product
** Frobenius norm
** Frobenius method
** Frobenius g ...
and
Thompson describe when a group has a
normal ''p''-complement.
Philip Hall characterized finite
soluble
In chemistry, solubility is the ability of a substance, the solute, to form a solution with another substance, the solvent. Insolubility is the opposite property, the inability of the solute to form such a solution.
The extent of the solubi ...
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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