A thin group is a finite group in which Sylow p -subgroups of 2-local subgroups are cyclic for any odd prime number p . Informally, these are groups that resemble a type 1 rank- type gupp over a finite field of characteristic 2.
Janko [1] defined thin groups and classified from them those that have characteristic type 2, in which all 2-local subgroups are solvable. Thin simple groups classified by Aschbacher [2] [3] . The list of finite simple thin groups consists of the following elements:
- Projective special linear groups , for ,
- Projective special unitary groups for and or 1,
- Sz (2 n )
- Tits Group 2 F 4 (2)
- Group Steinberg 3 D 4 (2)
- Mathieu Group M 11
See also
Notes
- ↑ Janko, 1972 .
- ↑ Aschbacher, 1976 .
- ↑ Aschbacher, 1978 .
Literature
- Michael Aschbacher. Thin finite simple groups // Bulletin of the American Mathematical Society . - 1976. - V. 82 , no. 3 - p . 484 . - ISSN 0002-9904 . - DOI : 10.1090 / S0002-9904-1976-14063-3 .
- Michael Aschbacher. Thin finite simple groups // Journal of Algebra . - 1978. - V. 54 , no. 1 . - p . 50–152 . - ISSN 0021-8693 . - DOI : 10.1016 / 0021-8693 (78) 90022-4 .
- Ashbakher M. Finite simple groups and their classification // UMN. - T. 36 , no. 2 (218) . - p . 141-172 .
- Zvonimir Janko. Nonsolvable finite groups 2-local subgroups are solvable. I // Journal of Algebra . - 1972. - T. 21 . - p . 458-517 . - ISSN 0021-8693 . - DOI : 10.1016 / 0021-8693 (72) 90009-9 .