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.
Contents
The transfer was defined by Issai Schur (1902) and rediscovered by Emil Artin (1929).
Construction
The construction of the map proceeds as follows: 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 yxi is in some coset xjH and so
for some index j and some element hi 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 hi 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}. 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 is finitely generated, the commutator subgroup G′ of G has finite index in G and H=G′, 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. See the Emil Artin-John Tate Class Field Theory notes.