{{Short description|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 N<sub>''G''</sub>(''Q'')/C<sub>''G''</sub>(''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 PSL<sub>2</sub>('''F'''<sub>7</sub>) 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 PSL<sub>2</sub>('''F'''<sub>7</sub>) 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 | access-date=2012-05-21 | archive-date=2011-08-07 | archive-url=https://web.archive.org/web/20110807060300/http://cms.math.ca/cjm/v20/p1101 | url-status=dead }} *{{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 }}
Category:Finite groups