# Transfer (group theory)

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In the mathematical field of group theory, the **transfer** defines, given a group *G* and a subgroup of finite index *H*, a group homomorphism from *G* to the abelianization of *H*. It can be used in conjunction with the Sylow theorems to obtain certain numerical results on the existence of finite simple groups.

The transfer was defined by Template:Harvs and rediscovered by Template:Harvs.^{[1]}

## Construction

The construction of the map proceeds as follows:^{[2]} Let [*G*:*H*] = *n* and select coset representatives, say

for *H* in *G*, so *G* can be written as a disjoint union

Given *y* in *G*, each *yx _{i}* is in some coset

*x*and so

_{j}Hfor some index *j* and some element *h*_{i} of *H*.
The value of the transfer for *y* is defined to be the image of the product

in *H*/*H*′, where *H*′ is the commutator subgroup of *H*. The order of the factors is irrelevant since *H*/*H*′ is abelian.

It is straightforward to show that, though the individual *h _{i}* depends on the choice of coset representatives, the value of the transfer does not. It's also straightforward to show that the mapping defined this way is a homomorphism.

## Example

If *G* is abelian then the transfer takes any element *y* of *G* to *y*^{[G:H]}.

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}.^{[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.^{[3]} 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 *H* is the commutator subgroup *G*′ of *G* and is finitely generated, 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.^{[1]} See the Emil Artin-John Tate *Class Field Theory* notes.

## See also

- Focal subgroup theorem, an important application of transfer
- By Artin's reciprocity law, the Artin transfer describes the principalization of ideal classes in extensions of algebraic number fields.

## References

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