Glauberman-Thompson normal p-complement theorem
From Groupprops
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 .
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
Contents
Statement
Direct statement
Suppose is a finite group and is an odd prime number. Let be a -Sylow subgroup. Then, if possesses a normal p-complement, so does . In other words, is a retract of .
In terms of functors and control of complements
For an odd prime , the ZJ-functor is a characteristic p-functor that controls normal p-complements in every finite group.
Related facts
- Burnside's normal p-complement theorem
- Frobenius' normal p-complement theorem
- Thompson's first normal p-complement theorem
- Thompson's second normal p-complement theorem
- Glauberman-Solomon normal p-complement theorem
Facts used
- Generalized Glauberman-Thompson normal p-complement theorem
- 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}