Alperin's fusion theorem in terms of conjugation families

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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:

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

References

Journal references

Textbook references

  • Book:GLMore info, Page 6-7, Theorems 3.4 and 3.5