Bad group
From Groupprops
This article defines a property of a group (possibly with additional operations and structure) as viewed in logic/model theory
History
Origin
This term was introduced by: Cherlin
The term bad group was introduced by Gregory Cherlin, as an example of groups that should not exist. They are groups that provide very strong counterexamples to the Cherlin conjecture.
Definition
Symbol-free definition
A group (possibly with additional structure and relations) is termed a bad group if it is connected, not solvable, has finite Morely rank, and in which every definable proper subgroup is nilpotent-by-finite. It is conjectured that bad groups do not exist.
References
- Groups of small Morley rank by Gregory Cherlin, Ann. Pure Appl. Logic 17 (1979), 1-28
- Solvable groups of finite Morely rank by Ali Nesin, Journal of Algebra 121 (1989), 26-39
- Groups of finite Morley rank without solvable non-nilpotent subgroups by Aleksandr Vasil'evic Borovic and Bruno Petrovic Poizat, Sibirskii Math Journal, 32 (2), 1991, 209