# 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. 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, that 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 PSL_{2}(**F**_{7}) 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 PSL_{2}(**F**_{7}) 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 - 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