# Normal p-complement
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{{Short description|Finite group}}
In [group theory](/source/group_theory), a branch of [mathematics](/source/mathematics), a '''normal ''p''-complement''' of a [finite](/source/finite_group) [group](/source/group_(mathematics)) for a [prime](/source/prime_number) ''p'' is a [normal subgroup](/source/normal_subgroup) of [order](/source/order_(group_theory)) [coprime](/source/coprime) to ''p'' and [index](/source/index_of_a_subgroup) a power of ''p''. In other words the group is a [semidirect product](/source/semidirect_product) of the normal ''p''-complement and any [Sylow ''p''-subgroup](/source/Sylow_subgroup). A group is called '''''p''-nilpotent''' if it has a normal {{nowrap|''p''-complement}}.
==Cayley normal 2-complement theorem==
[Cayley](/source/Arthur_Cayley) showed that if the Sylow 2-subgroup of a group ''G'' is [cyclic](/source/cyclic_group) then the group has a normal {{nowrap|2-complement}}, which shows that the Sylow {{nowrap|2-subgroup}} of a [simple group](/source/simple_group) of [even](/source/parity_(mathematics)) order cannot be cyclic.
==Burnside normal ''p''-complement theorem==
{{harvs|txt|year=1911|loc=Theorem II, section 243|last=[Burnside](/source/William_Burnside)}} showed that if a Sylow ''p''-subgroup of a group ''G'' is in the [center](/source/center_(group_theory)) of its [normalizer](/source/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](/source/trivial_group) 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''G''(''Q'')/C''G''(''Q'') is a [''p''-group](/source/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 subgroup](/source/characteristic_subgroup)s. For [odd](/source/parity_(mathematics)) 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](/source/simple_group) [PSL2('''F'''7)](/source/PSL(2%2C7)) of order 168 is a [counterexample](/source/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](/source/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](/source/Thompson_subgroup).
The result fails for ''p'' = 2 as the simple group [PSL2('''F'''7)](/source/PSL(2%2C7)) 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](/source/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](/source/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](/source/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](/source/Journal_of_Algebra) | issn=0021-8693 | volume=1 | pages=43–46| doi-access=free }}
Category:Finite groups
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Adapted from the Wikipedia article [Normal p-complement](https://en.wikipedia.org/wiki/Normal_p-complement) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Normal_p-complement?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.