# Alperin's fusion theorem in terms of conjugation families

## History

This statement was formulated and proved by Jonathan Lazare Alperin, in 1965.

## Statement

### General form

Suppose $G$ is a finite group, $S$ is a $p$-Sylow subgroup for a prime $p$ Suppose $\mathcal{F}$ is a collection of subgroups of $S$ with the property that for any subgroup $T$ of $S$, there exists $U \in \mathcal{F}$ such that:

• $U$ and $T$ are conjugate subgroups inside $G$
• $N_S(U)$ is a $p$-Sylow subgroup of $N_G(U)$

Then $\mathcal{F}$ is a conjugation family for $S$ in $G$.

### More specific form

This form states:

Let $\mathcal{F}$ be the collection of all subgroups of $S$ whose normalizer in $S$ is a Sylow subgroup of the normalizer in $G$. Then, $\mathcal{F}$ is a conjugation family for $S$ in $G$.

This follows from the more general form, and the fact that every p-subgroup is conjugate to a p-subgroup whose normalizer in the Sylow is Sylow in its normalizer.

## Proof

### Key idea: induction on index

We prove the result by inducting on the index of the subgroup $\langle A \rangle$ in $S$. Suppose $T = \langle A \rangle$ and $V = \langle B \rangle$.

### Base case for induction

The base case of induction is when $T = S$. By the conditions, $\mathcal{F}$ contains a conjugate of $S$, so $S \in \mathcal{F}$. Clearly, then $g \in N_G(S)$, so we can set $n = 1$, and $g_1 = g$.

### Induction step

The key thing to remember for the induction step is that if $T$ is a proper subgroup of $S$, then $T$ is a proper subgroup of $N_S(T)$ (and similarly for $V$). Thus, this step reduces to three parts:

• Go from $T$ to $N_S(T)$
• Use the induction to argue that we can go from $N_S(T)$ to $N_S(U)$
• Go from $N_S(V)$ back down to $V$