Transfer (group theory)
Encyclopedia
In mathematics
, the transfer in group theory
is a group homomorphism
defined given a group
G and a subgroup
of finite index H, which goes from the abelianization of G to that of H.
s of H in G, say
.
Given g in G, it is always possible to write
with some index j and some hi(g) in H; as one sees by asking which coset
is. The individual hi(g) depend on the choice made of coset representatives; but it turns out that the product
taken over all i is well-defined
, up to commutator
s in H. It also defines a homomorphism φ on G, again up to commutators and so into the abelianization of H. Finally this is a homomorphism from G to an abelian group
; it therefore is as good as a homomorphism ψ from the abelianisation of G to that of H. The mapping ψ is by definition the transfer from G to H.
on quadratic residues, which in effect computes the transfer for the multiplicative group of non-zero residue classes modulo a prime number
p, with respect to the subgroup {1, −1}. One advantage of looking at it that way is the ease with which the correct generalisation can be found, for example for cubic residues in the case that p − 1 is divisible by three.
(strictly, group homology), providing a more abstract definition. The transfer is also seen in algebraic topology
, when it is defined between classifying space
s of groups.
.
G′ of G, then the corresponding transfer map is trivial. In other words, the map sends G to 0 in the abelianization of G′. This is important in proving the principal ideal theorem
in class field theory
. See the Emil Artin
-John Tate
Class Field Theory notes.
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...
, the transfer in 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...
is a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
defined given 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 and 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...
of finite index H, which goes from the abelianization of G to that of H.
Formulation
To define the transfer, take coset representatives for the left cosetCoset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s of H in G, say
.Given g in G, it is always possible to write
with some index j and some hi(g) in H; as one sees by asking which coset
is. The individual hi(g) depend on the choice made of coset representatives; but it turns out that the product
- Π hi(g)
taken over all i is well-defined
Well-defined
In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...
, up to commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
s in H. It also defines a homomorphism φ on G, again up to commutators and so into the abelianization of H. Finally this is a homomorphism from G to an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
; it therefore is as good as a homomorphism ψ from the abelianisation of G to that of H. The mapping ψ is by definition the transfer from G to H.
Example
A simple case is that seen in the Gauss lemmaGauss's lemma (number theory)
Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity....
on quadratic residues, which in effect computes the transfer for the multiplicative group of non-zero residue classes modulo a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p, with respect to the subgroup {1, −1}. One advantage of looking at it that way is the ease with which the correct generalisation can be found, for example for cubic residues in the case that p − 1 is divisible by three.
Homological interpretation
This homomorphism may be set in the context of group cohomologyGroup cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
(strictly, group homology), providing a more abstract definition. The transfer is also seen in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, when it is defined between classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
s of groups.
Terminology
The name transfer translates the German Verlagerung, which was coined by Helmut HasseHelmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
.
Commutator subgroup
If H is the commutator subgroupCommutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group....
G′ of G, then the corresponding transfer map is trivial. In other words, the map sends G to 0 in the abelianization of G′. This is important in proving the principal ideal theorem
Principal ideal theorem
In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, is the statement that for any algebraic number field K and any ideal I of the ring of integers of K, if L is the Hilbert class field of K, thenIO_L\ is a principal ideal αOL, for OL the ring of...
in class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
. See the Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
-John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
Class Field Theory notes.