Groups of finite Morley rank: Glossary

• Bad field
A structure with K a field and T a subgroup of the multiplicative group of K is called a bad field if T is proper and infinite, and the structure has finite Morley rank. Work of Wagner and Poizat suggests that these should exist in characteristic 0, but not otherwise.
• Bad Group
Nonsolvable connected group of finite Morley rank with all connected definable sections nilpotent. Necessarily nonalgebraic. Sometimes assumed to be simple as well.
• Minimal connected simple
Connected simple, with all definable connected subgroups solvable.
Sometimes called minimal simple, but there is no assumption about finite sections.
• Normal 2-rank
The maximal rank of an elementary abelian 2-subgroup which is normal in a Sylow 2-subgroup.
• Prufer p-rank
The number of quasicyclic factors in a divisible abelian p-group A; can be calculated as the dimension of the socle A[p].
• Tame
A group G is tame in the strict sense if there is no bad field interpreted in it; in the narrow sense, if there is no bad field (K,T) such that K and T occur as definable sections, with the multiplication induced by conjugation in G. The narrow sense is sufficient.
Elsewhere in the literature "tame" includes another assumption, namely the noninvolvement of bad groups. But in the K* setting this is automatic (unless the group is itself bad), and outside the K* setting it is not appropriate.
• Torus
A torus is a definable abelian divisible group. A p-torus is the p-part of a torus; this will not be definable, if nontrivial.
• Unipotent
A group is unipotent if it is definable, connected, and of bounded exponent, and is p-unipotent if in addition it is a p-group. Solvable unipotent groups are nilpotent; all 2-unipotent groups are nilpotent. Sometimes the hypothesis of solvability is included in the definition.
Notions of unipotent radical differing from the above are useful as well. Cf. Burdges, dissertation, Rutgers 2004.

Reference: Borovik/Nesin, Groups of Finite Morley Rank, Oxford Logic Guids 26.

Groups of finite Morley rank