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 $F$ is a collection of subgroups of $S$ with the property that for any subgroup $T$ of $S$ , there exists $U \in F$ such that:

$U$ and $T$ are conjugate subgroups inside $G$
$N_{S} (U)$ is a $p$ -Sylow subgroup of $N_{G} (U)$

Then $F$ is a conjugation family for $S$ in $G$ .

More specific form

This form states:

Let $F$ be the collection of all subgroups of $S$ whose normalizer in $S$ is a Sylow subgroup of the normalizer in $G$ . Then, $F$ 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 $⟨ A ⟩$ in $S$ . Suppose $T = ⟨ A ⟩$ and $V = ⟨ B ⟩$ .

Base case for induction

The base case of induction is when $T = S$ . By the conditions, $F$ contains a conjugate of $S$ , so $S \in F$ . 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 $V$

References

Journal references

Transfer and fusion in finite groups by Jonathan Lazare Alperin and Daniel Gorenstein, Journal of Algebra, ISSN 00218693, Volume 6, Page 242 - 255(Year 1967): This paper discusses the normalizers of subgroups of a Sylow subgroup in a finite group, using the ideas of a conjugation family and Alperin's fusion theorem^{Weblink (hosted on ScienceDirect)}^{More info}

Textbook references

Book:GL^{More info}, Page 6-7, Theorems 3.4 and 3.5