# Transfer to an abelian group

## Definition

Let $G$ be a finite group and $H$ and $K$ be subgroups such that $K \triangleleft H$ and $H/K$ is abelian and $H$ has finite index in $G$. Let $T$ be a left transversal of $H$ in $G$. Then define the following mapping $V: G \to H/K$

$V(x) = \prod_{t \in T} h_x^T(t)K$

here $h_x^T(t)$ is the unique element $h \in H$ such that $xt = t'h$ for some $t' \in T$.

We need to quotient out by $K$ 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 $G$ to $H/K$.

### Independence of choice of transversal

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