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.
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[edit] Formulation
To define the transfer, take coset representatives for the left cosets 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, 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.
