Maximal among abelian normal subgroups of Sylow subgroup implies direct factor of centralizer

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Statement

Suppose G is a finite group and p is a prime number. Suppose P is a p-Sylow subgroup of G, and A is Maximal among abelian normal subgroups (?) in P. Then, A is a Direct factor (?) in C_G(A).

Related facts

Category:Normal p-complement theorems lists many theorems of a related nature.

Other related facts:

Facts used

  1. Maximal among abelian normal implies self-centralizing in nilpotent
  2. Product formula
  3. Burnside's normal p-complement theorem

Proof

Given: A finite group G, a prime p, a p-Sylow subgroup P of G. A subgroup A that is maximal among abelian normal subgroups of G.

To prove: A is normal in C_G(A). Further, there exists a subgroup B of G such that AB = C_G(A) and A \cap B is trivial.

Proof: Clearly, A</math, being abelian, is in the center of <math>C_G(A), hence normal in C_G(A).

  1. C_G(A) \cap P = C_P(A) = A: This follows from fact (1).
  2. PC_G(A) is a group: Since P normalizes A, P also normalizes C_G(A). Hence, PC_G(A) is a group.
  3. A is a Sylow subgroup of C_G(A): Consider the intersection C_G(A) \cap P. By the product formula (fact (2)), [C_G(A):(C_G(A) \cap P)] = [PC_G(A):P]. The right side is relatively prime to p, hence so is the left side. By step (1), C_G(A) \cap P = A, so [C_G(A):A] is coprime to p. Thus, A is a p-Sylow subgroup of C_G(A).
  4. A has a normal complement, say B, in C_G(A): By definition, A is in the center of C_G(A), so it is in the center of its normalizer in C_G(A), so by fact (3), A has a normal complement.

(4) completes the proof.

References

Textbook references

  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 259, Theorem 6.5, Chapter 7 (Fusion, transfer and p-factor groups) ,Section 7.6 (elementary applications), More info