- Statement
- References
In mathematics, the Walter theorem, proved by {{harvs|txt|last=Walter|first=John H.|authorlink=John H. Walter|year1=1967|year2=1969}}, describes the finite groups whose Sylow 2-subgroup is abelian. {{harvtxt|Bender|1970}} used Bender's method to give a simpler proof. StatementWalter's theorem states that if G is a finite group whose 2-sylow subgroups are abelian, then G/O(G) has a normal subgroup of odd index that is a product of groups each of which is a 2-group or one of the simple groups PSL2(q) for q = 2n or q = 3 or 5 mod 8, or the Janko group J1, or Ree groups 2G2(32n+1). 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. {{harvs|txt|last=Thompson|year1=1967|year2=1972|year3=1977}} and {{harvtxt|Bombieri|Odlyzko|Hunt|1980}} later showed that they are all Ree groups, and {{harvtxt|Enguehard|1986}} gave a unified exposition of this result. References- {{Citation | last1=Bender | first1=Helmut | title=On groups with abelian Sylow 2-subgroups | doi=10.1007/BF01109839 | mr=0288180 | year=1970 | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=117 | pages=164–176}}
- {{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | last2=Odlyzko | first2=Andrew | author2-link=Andrew Odlyzko | last3=Hunt | first3=D. | title=Thompson's problem (σ2=3) | doi=10.1007/BF01402275 | mr=570875 | year=1980 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=58 | issue=1 | pages=77–100}}
- {{Citation | last1=Enguehard | first1=Michel | title=Caractérisation des groupes de Ree | mr=873958 | year=1986 | journal=Astérisque | issn=0303-1179 | issue=142 | pages=49–139}}
- {{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Toward a characterization of E2(q) | doi=10.1016/0021-8693(67)90080-4 | mr=0223448 | year=1967 | journal=Journal of Algebra | issn=0021-8693 | volume=7 | pages=406–414}}
- {{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Toward a characterization of E2(q). II | doi=10.1016/0021-8693(72)90074-9 | mr=0313377 | year=1972 | journal=Journal of Algebra | issn=0021-8693 | volume=20 | pages=610–621}}
- {{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Toward a characterization of E2(q). III | doi=10.1016/0021-8693(77)90276-9 | mr=0453858 | year=1977 | journal=Journal of Algebra | issn=0021-8693 | volume=49 | issue=1 | pages=162–166}}
- {{Citation | last1=Walter | first1=John H.|authorlink=John H. Walter | title=Finite groups with abelian Sylow 2-subgroups of order 8 | doi=10.1007/BF01428899 | mr=0218445 | year=1967 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=2 | pages=332–376}}
- {{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 |series=Second Series | issn=0003-486X | volume=89 | pages=405–514 | doi=10.2307/1970648}}
2 : Finite groups|Theorems in group theory |