Characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it

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Statement

Suppose G is a finite group, p is a prime number, P is a p-Sylow subgroup (?), and K is a Characteristic subgroup (?) of P. Then, K is a Weakly closed subgroup (?) in P (relative to G) if and only if K is a normal subgroup in every p-Sylow subgroup containing it.

Related facts

p-functor version

Facts used

  1. Weakly closed implies conjugation-invariantly relatively normal in finite group
  2. Sylow implies order-conjugate: For a prime p, any two p-Sylow subgroups are conjugate.
  3. Sylow implies WNSCDIN
  4. WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

Proof

Given: A finite group G, a prime p, a p-Sylow subgroup P of G. A characteristic subgroup K of P.

Weakly closed implies normal in every Sylow subgroup containing it

To prove: Given that K is weakly closed in P, and Q is a p-Sylow subgroup of G containing K, K is normal in Q.

Proof (quick version): This follows from facts (1) and (2). By fact (2), P and Q are conjugate, and by fact (1), K is normal in Q. (Note that this part does not use the assumption that K is a characteristic subgroup of P).

Proof (hands-on): PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Normal in every Sylow subgroup containing it implies weakly closed

To prove: Given that K is normal in every p-Sylow subgroup of G containing it, K is weakly closed in P. In other words, if g \in G is such that gKg^{-1} \le P, then gKg^{-1} = K.

Proof (quick version): The proof follows from facts (3) and (4), and the observation (again stemming from fact (2)) that the conjugates of a particular p-Sylow subgroup are precisely all the p-Sylow subgroups.

Proof (hands-on): PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

References

Textbook references

  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 255, Theorem 5.1, Chapter 7 (Fusion, transfer and p-factor groups), Section 7.5 (Weak closure and p-normality), More info