Transfer (group theory)

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 Issai Schur (1902) and rediscovered by Emil Artin (1929).

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 yx_i is in some coset x_jH and so

for 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.

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.

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