Transfer homomorphism

Summary

The transfer is a group homomorphism from a group $G$ to the abelianisation of a subgroup $H$ of finite index. It is built from the way $G$ permutes the left cosets of $H$. The transfer gives normal subgroups and abelian quotients of $G$ from information about $H$, usually a Sylow subgroup. It proves results such as Burnside's normal $p$-complement theorem and the focal subgroup theorem.