In the mathematical

classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...

, a thin group is a finite group such that for every odd

prime number 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'', the Sylow ''p''-subgroups of the 2- local subgroups are

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 ...

. Informally, these are the groups that resemble rank 1

groups of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phra ...

over a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

of characteristic 2. defined thin groups and classified those of characteristic 2 type in which all 2-local subgroups are solvable. The thin

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 were classified by . The list of finite simple thin groups consists of: *The projective special linear groups PSL₂(''q'') and PSL₃(''p'') for ''p'' = 1 + 2^''a''3^''b'' and PSL₃(4) *The projective special unitary groups PSU₃(''p'') for ''p'' =−1 + 2^''a''3^''b'' and ''b'' = 0 or 1 and PSU₃(2^''n'') *The

Suzuki groups In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22''n''+1), 2''B''2(22''n''+1), Suz(22''n''+1), or ''G''(22''n''+1), form an infinite family of groups of Lie type found by , that are simple for ''n'' ≥ 1. ...

Sz(2^''n'') *The

Tits group In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order : 211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group ...

²''F''₄(2)' *The Steinberg group ³''D''₄(2) *The

Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...

''M''₁₁ *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 : 233571119 = 175560 : ≈ 2. History ''J1'' is one of the 26 sporadic groups and was originally described by Zvo ...

References

* * *{{Citation , last1=Janko , first1=Zvonimir , title=Nonsolvable finite groups all of whose 2-local subgroups are solvable. I , doi=10.1016/0021-8693(72)90009-9 , mr=0357584 , year=1972 , 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. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to ...

, issn=0021-8693 , volume=21 , pages=458–517, doi-access=free Finite groups

See also

References