---
tag: 3R2Y906QAQ
kind: concept
name: Transfer homomorphism
msc: [20D20, 20J06]
specializes: [0JYTG5K9Y4]
sources:
  - url: https://en.wikipedia.org/wiki/Transfer_(group_theory)
same_as:
  wikidata: Q3537483
  wikipedia: Transfer (group theory)
---

## Summary

The transfer is a [[group homomorphism|0JYTG5K9Y4]] from a group $G$ to the [[abelianisation|0AQZV0905N]] of a subgroup $H$ of finite [[index|0DTXBQPA2S]].
It is built from the way $G$ permutes the [[left cosets|03DEH43F4H]] of $H$.
The transfer gives normal subgroups and abelian quotients of $G$ from information about $H$, usually a [[Sylow subgroup|0053CQE9Y2]].
It proves results such as Burnside's normal $p$-complement theorem and the [[focal subgroup theorem|H1BSG0J5G1]].

## Explanations

### L1 human

Let $H$ be a subgroup of $G$ with $n$ cosets.
An element $g$ of $G$ shuffles the cosets, and each step of the shuffle leaves a small correction term in $H$.
The transfer multiplies these $n$ correction terms together, in a version of $H$ where the order of multiplication does not matter.
When $G$ is abelian, the result is simply $g^n$.
For the integers with $H$ the multiples of $n$, the transfer sends $1$ to $n$.