Difference between revisions of "Alperin's fusion theorem in terms of conjugation families"

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* <math>N_S(U)</math> is a <math>p</math>-[[Sylow subgroup]] of <math>N_G(U)</math>
 
* <math>N_S(U)</math> is a <math>p</math>-[[Sylow subgroup]] of <math>N_G(U)</math>
  
Then <math>\mathcal{F}</math> is a conjugation family for <math>S</math> in <math>G</math>.
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Then <math>\mathcal{F}</math> is a [[conjugation family]] for <math>S</math> in <math>G</math>.
  
 
===More specific form===
 
===More specific form===
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This form states:
 
This form states:
  
Let <math>\mathcal{F}</math> be the collection of ''all '' subgroups of <math>S</math> whose normalizer in <math>S</math> is a Sylow subgroup of the normalizer in <math>G</math>. Then, <math>\mathcal{F}</math> is a conjugation family for <math>S</math> in <math>G</math>.
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Let <math>\mathcal{F}</math> be the collection of ''all '' subgroups of <math>S</math> whose normalizer in <math>S</math> is a Sylow subgroup of the normalizer in <math>G</math>. Then, <math>\mathcal{F}</math> is a [[conjugation family]] for <math>S</math> in <math>G</math>.
  
 
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]].
 
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]].

Revision as of 14:46, 17 June 2008

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