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 "h_i"("g") in "H"; as one sees by asking which coset

:$gcdot\{g\}\_iH$

is. The individual "h_i"("g") depend on the choice made of coset representatives; but it turns out that the product

:Π "h_i"("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