View source for Group of GF(2)-type

In mathematical finite group theory, a '''group of GF(2)-type''' is a group with an involution centralizer whose [[Generalized Fitting subgroup|generalized Fitting subgroup]] is a [[Group of symplectic type|group of symplectic type]] {{harv|Gorenstein|1982|loc=definition 1.45}}.
As the name suggests, many of the [[Group of Lie type|groups of Lie type]] over the field with 2 elements are groups of GF(2)-type. Also 16 of the 26 sporadic groups are of GF(2)-type, suggesting that in some sense sporadic groups are somehow related to special properties of the field with 2 elements.

{{harvtxt|Timmesfeld|1978}} showed roughly that groups of GF(2)-type can be subdivided into 8 types. The groups of each of these 8 types were classified by various authors. They consist mainly of groups of Lie type with all roots of the same length over the field with 2 elements, but also include many exceptional cases, including the majority of the sporadic simple groups. 
{{harvtxt|Smith|1980}} gave a survey of this work.

{{harvtxt|Smith|1979|loc=p.279}} gives a table of simple groups  containing a large extraspecial 2-group.

==References==
*{{Citation | last1=Gorenstein | first1=D. |  title=Finite simple groups | publisher=Plenum Publishing Corp. | location=New York | series=University Series in Mathematics | isbn=978-0-306-40779-6 | mr=698782 | year=1982 | url-access=registration | url=https://archive.org/details/finitesimplegrou0000gore }}
*{{Citation | last1=Smith | first1=Stephen D. | title=Large extraspecial subgroups of widths 4 and 6 | doi=10.1016/0021-8693(79)90160-1 | mr=540638 | year=1979 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=58 | issue=2 | pages=251–281| doi-access=free }}
*{{Citation | last1=Smith | first1=Stephen D. | title=The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) | url=https://books.google.com/books?isbn=0821814400 | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Proc. Sympos. Pure Math. | mr=604567 | year=1980 | volume=37 | chapter=The classification of finite groups with large extraspecial 2-subgroups | pages=111–120}}
*{{Citation | last1=Timmesfeld | first1=Franz | title=Finite simple groups in which the generalized Fitting group of the centralizer of some involution is extraspecial | doi=10.2307/1971146 | mr=486255 | year=1978 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=107 | issue=2 | pages=297–369}}  Correction: {{Citation | last1=Timmesfeld | first1=Franz | title=Correction to Finite simple groups in which the generalized Fitting group of the centralizer of some involution is extraspecial | jstor=1971119 | mr=486255 | year=1979 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=109 | issue=2 | pages=413–414 | doi=10.2307/1971119}}

[[Category:Finite groups]]



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