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
This page © T. Altinel, A. Borovik, and G. Cherlin.
Errors and omissions © G. Cherlin.