group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a normal ''p''-complement of a

finite Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

for a

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

''p'' is a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...

to ''p'' and

index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...

a power of ''p''. In other words the group is a

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...

of the normal ''p''-complement and any Sylow ''p''-subgroup. A group is called ''p''-nilpotent if it has a normal .

Cayley normal 2-complement theorem

Cayley showed that if the Sylow 2-subgroup of a group ''G'' is

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...

then the group has a normal , which shows that the Sylow of a

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname), a Breton surname * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a ...

order cannot be cyclic.

Burnside normal ''p''-complement theorem

showed that if a Sylow ''p''-subgroup of a group ''G'' is in the

center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...

of its

normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

then ''G'' has a normal . This implies that if ''p'' is the smallest prime dividing the order of a group ''G'' and the Sylow is cyclic, then ''G'' has a normal .

Frobenius normal ''p''-complement theorem

The Frobenius normal ''p''-complement theorem is a strengthening of the Burnside normal theorem, which states that if the normalizer of every

non-trivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group, topological space). The noun triviality usual ...

subgroup of a Sylow of ''G'' has a normal , 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 has a normal 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 , one uses only the non-trivial

characteristic subgroup In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphis ...

s. For odd primes ''p'' Thompson found such a strengthened criterion: in fact he did not need all characteristic subgroups, but only two special ones. showed that if ''p'' is an odd prime and the groups N(J(''P'')) and C(Z(''P'')) both have normal for a Sylow of ''G'', then ''G'' has a normal . In particular if the normalizer of every nontrivial characteristic subgroup of ''P'' has a normal , then so does ''G''. This consequence is sufficient for many applications. The result fails for ''p'' = 2 as the

PSL₂(F₇) of order 168 is a

counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...

. 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 . Glauberman improved this further by showing that one only needs to use one characteristic subgroup: the center of the Thompson subgroup. used his

ZJ theorem In mathematics, George Glauberman's ZJ theorem states that if a finite group ''G'' is ''p''-constrained and ''p''-stable and has a normal ''p''-subgroup for some odd prime ''p'', then ''O(''G'')''Z''(''J''(''S'')) is a normal subgroup of ''G ...

to prove a normal theorem, that if ''p'' is an odd prime and the normalizer of Z(J(P)) has a normal , for ''P'' a Sylow 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

* Reprinted by Dover 1955 * * * *{{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 ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier Elsevier ( ) is a Dutch academic publishing company specializing in scientific, te ...

, issn=0021-8693 , volume=1 , pages=43–46, doi-access=free Finite groups