Alperin's fusion theorem in terms of conjugation families

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

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.

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 $$V$$

Textbook references

 * , Page 6-7, Theorems 3.4 and 3.5