---
tag: H1BSG0J5G1
kind: statement
role: theorem
name: Focal subgroup theorem
msc: [20D20]
uses: [G5S1M9, 01826KMG2H, 0053CQE9Y2, 0ASGSTXSBW, 0DXD4QSZ8A, 0RP7VWQ0YN, 3R2Y906QAQ, 0AQZV0905N]
about: [0053CQE9Y2, 0ASGSTXSBW]
sources:
  - url: https://en.wikipedia.org/wiki/Focal_subgroup_theorem
same_as:
  wikidata: Q5463860
  wikipedia: Focal subgroup theorem
---

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a [[prime number|01826KMG2H]], and let $P$ be a [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$.
Let $G'$ be the [[commutator subgroup|0ASGSTXSBW]] of $G$.
The *focal subgroup* of $P$ in $G$ is [[generated|0RP7VWQ0YN]] by all $x^{-1} y$ with $x, y \in P$ and $y$ [[conjugate|0DXD4QSZ8A]] to $x$ in $G$:
$$\operatorname{Foc}_G(P) = \langle x^{-1} y : x, y \in P, \ y = g^{-1} x g \text{ for some } g \in G \rangle.$$
Then
$$P \cap G' = \operatorname{Foc}_G(P).$$

## Remarks

The inclusion $\operatorname{Foc}_G(P) \subseteq P \cap G'$ is easy, because $x^{-1} g^{-1} x g$ is a commutator.
The other inclusion uses the [[transfer|3R2Y906QAQ]] $V \colon G \to P/P'$ and a formula for $V(x)$ with $x \in P$ in terms of conjugates of powers of $x$ that lie in $P$.

Example: $G = S_3$.
For $p = 3$, $P = A_3$, and $(1\,2\,3)$ is conjugate to $(1\,3\,2)$, so $\operatorname{Foc}_G(P)$ contains $(1\,2\,3)^{-1}(1\,3\,2) = (1\,2\,3)$ and equals $A_3 = P \cap G'$.
For $p = 2$, $P = \set{e, (1\,2)}$, the only element of $P$ conjugate to $(1\,2)$ is $(1\,2)$ itself, so $\operatorname{Foc}_G(P) = \set{e}$; indeed $P \cap A_3 = \set{e}$.

As a consequence, the largest abelian $p$-group that is a quotient of $G$ is isomorphic to $P / \operatorname{Foc}_G(P)$.
So the abelian $p$-quotients of $G$ can be read off from $P$ and the way $G$ conjugates the elements of $P$.

## Explanations

### L1 human

The commutator subgroup measures how far a group is from being abelian.
The theorem says that its part inside a Sylow subgroup $P$ is generated by the "jumps" between elements of $P$ that the whole group makes conjugate.
So a question about the whole group reduces to a question about $P$ and conjugation.

### L2 human

The focal subgroup theorem is a central result of transfer theory.
It tells when a finite group has a nontrivial abelian $p$-group as a quotient: exactly when $\operatorname{Foc}_G(P) \ne P$.
Like Burnside's normal $p$-complement theorem, it is proved with the transfer, and it leads to the study of fusion, that is, of which elements of $P$ are conjugate in $G$.
The [[Z* theorem|2XCJ8NNMXP]] is another result about fusion, for involutions.