Complement (group theory)
Encyclopedia
In mathematics
, especially in the area of algebra
known as group theory
, a complement of a subgroup
H in a group
G is a subgroup K of G such that G = HK = { hk : h ∈ H and k ∈ K } and H ∩ K = {e}, that is, if every element of G has a unique expression as a product hk where h in H and k in K. Complements generalize both the direct product
(where the subgroups H and K commute element-wise), and the semidirect product
(where one of H or K normalizes the other). The product corresponding to a general complement is called the Zappa-Szép product
. In all cases, a subgroup with a complement, in some sense, lets the group be factored into simpler pieces.
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 group
s as those with p-complements for every 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
is one where every subgroup has a complement.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the area of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
known as group theory
Group theory
In mathematics and 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 groups endowed with additional operations and...
, a complement of a subgroup
Subgroup
In group theory, 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 subgroup of G if the restriction of * to H x H is a group operation on H...
H in a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G is a subgroup K of G such that G = HK = { hk : h ∈ H and k ∈ K } and H ∩ K = {e}, that is, if every element of G has a unique expression as a product hk where h in H and k in K. Complements generalize both the direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
(where the subgroups H and K commute element-wise), and the semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
(where one of H or K normalizes the other). The product corresponding to a general complement is called the Zappa-Szép product
Zappa-Szep product
In mathematics, especially group theory, the Zappa–Szép product describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products...
. In all cases, a subgroup with a complement, in some sense, lets the group be factored into simpler pieces.
A p-complement is a complement to a Sylow p-subgroup. Theorems of Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
and Thompson
John G. Thompson
John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize....
describe when a group has a normal p-complement. Philip Hall
Philip Hall
Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...
characterized finite soluble
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
groups amongst finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s as those with p-complements for every 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
Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point.They are named after F. G. Frobenius.- Structure :...
.
A complemented group
Complemented group
In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways.In , a complemented group is one in which every subgroup has a group-theoretic complement...
is one where every subgroup has a complement.