Normal p-complement#Glauberman normal p-complement theorem
{{Short description|A finite group}}
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 {{nowrap|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 {{nowrap|2-complement}}, which shows that the Sylow {{nowrap|2-subgroup}} of a simple group of even order cannot be cyclic.
Burnside normal ''p''-complement theorem
{{harvs|txt|year=1911|loc=Theorem II, section 243|last=Burnside}} showed that if a Sylow p-subgroup of a group G is in the center of its normalizer then G has a normal {{nowrap|p-complement}}. This implies that if p is the smallest prime dividing the order of a group G and the Sylow {{nowrap|p-subgroup}} is cyclic, then G has a normal {{nowrap|p-complement}}.
Frobenius normal ''p''-complement theorem
The Frobenius normal p-complement theorem is a strengthening of the Burnside normal {{nowrap|p-complement}} theorem, which states that if the normalizer of every non-trivial subgroup of a Sylow {{nowrap|p-subgroup}} of G has a normal {{nowrap|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 NG(Q)/CG(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 {{nowrap|p-subgroup}} has a normal {{nowrap|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 {{nowrap|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.
{{harvtxt|Thompson|1964}} showed that if p is an odd prime and the groups N(J(P)) and C(Z(P)) both have normal {{nowrap|p-complements}} for a Sylow {{nowrap|P-subgroup}} of G, then G has a normal {{nowrap|p-complement}}.
In particular if the normalizer of every nontrivial characteristic subgroup of P has a normal {{nowrap|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.
{{harvtxt|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 {{nowrap|p-subgroup}}. Glauberman improved this further by showing that one only needs to use one characteristic subgroup: the center of the Thompson subgroup.
{{harvtxt|Glauberman|1968}} used his ZJ theorem to prove a normal {{nowrap|p-complement}} theorem, that if p is an odd prime and the normalizer of Z(J(P)) has a normal {{nowrap|p-complement}}, for P a Sylow {{nowrap|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
- {{Citation | last1=Burnside | first1=William | author1-link=William Burnside | title=Theory of groups of finite order | orig-year=1897 | url=https://archive.org/details/theorygroupsfin00burngoog | publisher=Cambridge University Press | edition=2nd | isbn=978-1-108-05032-6 |mr=0069818 | year=1911}} Reprinted by Dover 1955
- {{Citation | last1=Glauberman | first1=George | author1-link=George Glauberman | title=A characteristic subgroup of a p-stable group | url=http://www.cms.math.ca/cjm/v20/p1101 |mr=0230807 | year=1968 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=20 | pages=1101–1135 | doi=10.4153/cjm-1968-107-2| doi-access=free }}
- {{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite groups | url=https://www.ams.org/bookstore-getitem/item=CHEL-301-H | publisher=Chelsea Publishing Co. | location=New York | edition=2nd | isbn=978-0-8284-0301-6 |mr=569209 | year=1980}}
- {{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Normal p-complements for finite groups | doi=10.1007/BF01162958 | mr=0117289 | year=1960 | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=72 | pages=332–354| s2cid=120848984 }}
- {{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Normal p-complements for finite groups | doi=10.1016/0021-8693(64)90006-7 |mr=0167521 | year=1964 | journal=Journal of Algebra | issn=0021-8693 | volume=1 | pages=43–46| doi-access=free }}