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 ...

, the Walter theorem, proved by , describes the

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 ...

s whose

Sylow 2-subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...

abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group ...

. used

Bender's method In group theory, Bender's method is a method introduced by for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify the Walter theorem on groups with abelian Sylow 2-subgroups , and G ...

to give a simpler proof.

Statement

Walter's theorem states that if ''G'' is a finite group whose 2-Sylow subgroups are abelian, then ''G''/''O''(''G'') has 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 ...

of odd

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 ...

that is a product of groups each of which is a

2-group In mathematics, particularly category theory, a is a groupoid with a way to multiply objects and morphisms, making it resemble a group. They are part of a larger hierarchy of . They were introduced by Hoàng Xuân Sính in the late 1960s unde ...

or one of the

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 ...

s PSL₂(''q'') for ''q'' = 2^''n'' or ''q'' = 3 or 5

mod Mod, MOD or mods may refer to: Places * Modesto City–County Airport, Stanislaus County, California, US Arts, entertainment, and media Music * Mods (band), a Norwegian rock band * M.O.D. (Method of Destruction), a band from New York City, US * ...

8, or the

Janko group J1 In the area of modern algebra known as group theory, the Janko group ''J1'' is a sporadic simple group of Order (group theory), order : 175,560 = 233571119 : ≈ 2. History ''J1'' is one of the 26 sporadic groups and was origi ...

, or

Ree group In mathematics, a Ree group is a group of Lie type over a finite field, finite field (mathematics), field constructed by from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki ...

s ²''G''₂(3^2''n''+1). (Here ''O''(''G'') denotes the unique largest normal subgroup of ''G'' of odd order.) The original statement of Walter's theorem did not quite identify the Ree groups, but only stated that the corresponding groups have a similar subgroup structure as Ree groups. and later showed that they are all Ree groups, and gave a unified exposition of this result.

References

* * * * * * * *{{Citation , last1=Walter , first1=John H., authorlink=John H. Walter , title=The characterization of finite groups with abelian Sylow 2-subgroups. , jstor=1970648 , mr=0249504 , year=1969 , journal=

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

, series=Second Series , issn=0003-486X , volume=89 , pages=405–514 , doi=10.2307/1970648 Theorems about finite groups