Normal p-complement

In group theory, a branch of mathematics, 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. A group is called p-nilpotent if it has a normal p-complement.

Cayley normal 2-complement theorem

Cayley showed that if the Sylow 2-subgroup of a group G is cyclic then the group has a normal 2-complement, which shows that the Sylow 2-subgroup of a simple group of even order cannot be cyclic.

Burnside normal p-complement theorem

Burnside (1911, Theorem II, section 243) showed that if a Sylow p-subgroup of a group G is in the center of its normalizer then G has a normal p-complement. This implies 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.

Frobenius normal p-complement theorem

The Frobenius normal p-complement theorem is a strengthening of the Burnside normal p-complement theorem, which 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. More precisely, the following conditions are equivalent:

G has a normal p-complement
The normalizer of every non-trivial p-subgroup has a normal p-complement
For every p-subgroup Q, the group N_G(Q)/C_G(Q) is a p-group.

Thompson normal p-complement theorem

The Frobenius normal p-complement theorem shows that if every normalizer of a non-trivial subgroup of a Sylow p-subgroup has a normal p-complement then so does G. For applications it is often useful to have a stronger version where instead of using all non-trivial subgroups of a Sylow p-subgroup, one uses only the non-trivial characteristic subgroups. For odd primes p Thompson found such a strengthened criterion: in fact he did not need all characteristic subgroups, but only two special ones.

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 PSL₂(F₇) of order 168 is a counterexample.

Thompson (1960) gave a weaker version of this theorem.

Glauberman normal p-complement theorem

Thompson's normal p-complement theorem used conditions on two particular characteristic subgroups of a Sylow p-subgroup. Glauberman improved this further by showing that one only needs to use one characteristic subgroup: the center of the Thompson subgroup.

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 result fails for p = 2 as the simple group PSL₂(F₇) of order 168 is a counterexample.

References

Burnside, William (1911) [1897], Theory of groups of finite order (2nd ed.), Cambridge University Press, ISBN 978-1-108-05032-6, MR 0069818 Reprinted by Dover 1955
Glauberman, George (1968), "A characteristic subgroup of a p-stable group", Canadian Journal of Mathematics, 20: 1101–1135, doi:10.4153/cjm-1968-107-2, ISSN 0008-414X, MR 0230807
Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR 0569209
Thompson, John G. (1960), "Normal p-complements for finite groups", Mathematische Zeitschrift, 72: 332–354, doi:10.1007/BF01162958, ISSN 0025-5874, MR 0117289, S2CID 120848984
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, MR 0167521