Transfer to an abelian group

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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.