Normal p-complement
In mathematical group theory, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power of p. In other words the group is a semidirect product of the normal p-complement and any Sylow p-subgroup.
Burnside normal p-complement theorem
Burnside showed that if p is the smallest prime dividing the order of a group G and the Sylow p-subgroup is cyclic, then G has a normal p-complement. In particular if the Sylow 2-subgroup is cyclic then G has a normal 2-complement, which shows that the Sylow 2-subgroup of a simple group of even order cannot be cyclic.
Frobenius normal p-complement theorem
The Frobenius normal p-complement theorem states that if the normalizer of every non-trivial subgroup of a Sylow p-subgroup of G has a normal p-complement, then so does G.
Thompson normal p-complement theorem
Thompson (1964) showed that if p is an odd prime and the groups N(J(P)) and C(Z(P) both have normal p-complements for a Sylow P-subgroup of G, then G has a normal p-complement.
In particular if the normalizer of every nontrivial characteristic subgroup of P has a normal p-complement, then so does G. This consequence is sufficient for many applications.
The result fails for p = 2 as the simple group of order 168 is a counterexample.
Thompson (1960) gave a weaker version of this theorem.
Glauberman normal p-complement theorem
Glauberman (1968) used his ZJ theorem to prove a normal p-complement theorem, that if p is an odd prime and the normalizer of Z(J(P)) has a normal p-complement, for P a Sylow p-subgroup of G, then so does G. Here Z stands for the center of a group and J for the Thompson subgroup.
The theorem fails for p=2: the simple group PSL2(F7) of order 168 is a counterexample.
References
- Glauberman, George (1968), "A characteristic subgroup of a p-stable group", Canadian Journal of Mathematics, 20: 1101–1135, ISSN 0008-414X, MR0230807
- Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR569209
- Thompson, John G. (1960), "Normal p-complements for finite groups", Mathematische Zeitschrift, 72: 332–354, doi:10.1007/BF01162958, ISSN 0025-5874, MR 0117289
- Thompson, John G. (1964), "Normal p-complements for finite groups", Journal of Algebra, 1: 43–46, doi:10.1016/0021-8693(64)90006-7, ISSN 0021-8693, MR0167521