Transfer (group theory)

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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.

1 Formulation
2 Example
3 Homological interpretation
4 Terminology
5 Commutator subgroup

[edit] 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

$g\cdot{g}_i = g_j\cdot{h}_i(g)$

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

$g\cdot{g}_iH$

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.

[edit] 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.

[edit] 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.

[edit] Terminology

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

[edit] Commutator subgroup

If G has commutator subgroup G′, then the corresponding transfer map is trivial, and sends G′ abelianized to 0 in G abelianized. This is important in proving the principal ideal theorem in class field theory. See the Emil Artin-John Tate Class Field Theory notes.