Transfer (group theory)

Transfer (group theory)

In mathematics, the transfer in group theory is a group homomorphism defined given a finite group "G" and a subgroup "H", which goes from the abelianization of "G" to that of "H".

Formulation

To define the transfer, take coset representatives for the left cosets of "H" in "G", say

:g_1, ldots, g_k.

Given "g" in "G", it is always possible to write

:gcdot{g}_i = g_jcdot{h}_i(g)

with some index "j" and some "hi"("g") in "H"; as one sees by asking which coset

:gcdot{g}_iH

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, up to commutators 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".

Example

A simple case is that seen in the Gauss lemma 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.

Homological interpretation

This homomorphism may be set in the context of group cohomology (strictly, group "homology"), providing a more abstract definition. The transfer is also seen in algebraic topology, when it is defined between classifying spaces of groups.

Terminology

The name "transfer" translates the German "Verlagerung", which was coined by Helmut Hasse.

Commutator subgroup

If "G" has commutator subgroup "G"′, then the corresponding transfer map is trivial, that is, it 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.

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Transfer sales theory — Importance of Transfer sales theory (TST) is critical for any commercial organization. Expanding business is not possible without increasing sales volumes, and effective sales management goal is to organize sales team work in such a manner that… …   Wikipedia

  • List of group theory topics — Contents 1 Structures and operations 2 Basic properties of groups 2.1 Group homomorphisms 3 Basic types of groups …   Wikipedia

  • Transfer — may refer to:*Call transfer *Decal *Electron transfer *Jacoby transfer, a bidding device in contract bridge *Knowledge transfer *Language transfer, where native language grammar and pronunciation influence the learning and use of a second… …   Wikipedia

  • Transfer pricing — refers to the pricing of contributions (assets, tangible and intangible, services, and funds) transferred within an organization. For example, goods from the production division may be sold to the marketing division, or goods from a parent… …   Wikipedia

  • Group delay and phase delay — Group delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component. Phase delay is a similar measure of the time… …   Wikipedia

  • Group affective tone — represents the consistent or homogeneous affective reactions within a group5,6. Group affective tone is an aggregate of the moods of the individual members of the group and refers to mood at the group level of analysis. If the moods of the… …   Wikipedia

  • Transfer function — A transfer function (also known as the system function[1] or network function) is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time invariant system. With… …   Wikipedia

  • Renormalization group — In theoretical physics, the renormalization group (RG) refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales. In particle physics, it reflects the… …   Wikipedia

  • Gauss's lemma (number theory) — This article is about Gauss s lemma in number theory. Gauss s lemma (polynomial) concerns factoring polynomials. Gauss s lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally …   Wikipedia

  • training, transfer of — In psychology, the effect of having learned one activity on an individual s execution of other activities. Positive transfer occurs when a previously acquired skill enhances one s performance of a new one. Negative transfer occurs when the… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”