# Transfer to an abelian group

From Groupprops

## Definition

Let be a finite group and and be subgroups such that and is abelian and has finite index in . Let be a left transversal of in . Then define the following mapping

here is the unique element such that for some .

We need to quotient out by so that the product on the right side is independent of the order of terms in the transversal.

## Facts

### Homomorphism

The transfer is a homomorphism of groups from to .

### Independence of choice of transversal

The transfer map does not depend on the choice of transversal .