Focal subgroup theorem

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 $V \colon G \to P/P'$ 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 $P$.

Going deeper

For a student

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 is another result about fusion, for involutions.