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 ...

, George Glauberman's ZJ theorem states that if a

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

''G'' is ''p''-constrained and ''p''-stable and has a normal ''p''-subgroup for some odd

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'', then ''O''_'(''G'')''Z''(''J''(''S'')) is a normal

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

of ''G'', for any Sylow ''p''-subgroup ''S''.

Notation and definitions

*''J''(''S'') is the Thompson subgroup of a ''p''-group ''S'': the subgroup generated by the abelian subgroups of maximal

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 ...

. *''Z''(''H'') means the center of a group ''H''. *''O''_' is the maximal normal subgroup of ''G'' of order

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'', the '-core *''O''_''p'' is the maximal normal ''p''-subgroup of ''G'', the ''p''-core. *''O''_',''p''(''G'') is the maximal normal ''p''-nilpotent subgroup of ''G'', the ',''p''-core, part of the upper ''p''-series. *For an odd prime ''p'', a group ''G'' with ''O''_''p''(''G'') ≠ 1 is said to be ''p''-stable if whenever ''P'' is a of ''G'' such that ''PO''(''G'') is normal in ''G'', and 'P'',''x'',''x''= 1, then the image of ''x'' in N_''G''(''P'')/C_''G''(''P'') is contained in a normal of N_''G''(''P'')/C_''G''(''P''). *For an odd prime ''p'', a group ''G'' with ''O''_''p''(''G'') ≠ 1 is said to be ''p''-constrained if the

centralizer 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 ...

C_''G''(''P'') is contained in ''O''_',''p''(''G'') whenever ''P'' is a Sylow of ''O''_',''p''(''G'').

References

* * * Theorems about finite groups {{Abstract-algebra-stub