Glauberman-Thompson normal p-complement theorem

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This article gives the statement, and possibly proof, of a normal p-complement theorem: necessary and/or sufficient conditions for the existence of a Normal p-complement (?). In other words, it gives necessary and/or sufficient conditions for a given finite group to be a P-nilpotent group (?) for some prime number p.
View other normal p-complement theorems
This article states a fact about the behavior of a finite group relative to a prime number. This fact is true only for odd primes, i.e., it breaks down for the prime two.
View similar facts

Statement

Direct statement

Suppose G is a finite group and p is an odd prime number. Let P be a p-Sylow subgroup. Then, if N_G(Z(J(P)) possesses a normal p-complement, so does G. In other words, P is a retract of G.

In terms of functors and control of complements

For an odd prime p, the ZJ-functor is a characteristic p-functor that controls normal p-complements in every finite group.

Related facts

Facts used

  1. Generalized Glauberman-Thompson normal p-complement theorem
  2. Strongly p-solvable implies Glauberman type for odd p: This states that in a strongly p-solvable group, the ZJ-subgroup functor is a characteristic p-functor whose normalizer generates whole group with p'-core. In particular, for a p'-core-free strongly p-solvable group, the ZJ-subgroup of any p-Sylow subgroup is normal (in fact, characteristic) in the whole group. See also group of Glauberman type for a prime.

Proof

The proof follows directly from Facts (1) and (2). Fact (2) basically says that the ZJ-subgroup functor satisfies the necessary conditions for us to be able to apply Fact (1).

References

Textbook references

  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 280, Theorem 3.1, Chapter 8 (p-constrained and p-stable groups), More info