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Comparing quasi-finitely axiomatizable and prime groups
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Andre Nies
Published/Copyright:
June 18, 2007
Abstract
An infinite f.g. group G is quasi-finitely axiomatizable (QFA) if there is a first-order sentence ϕ such that G ⊨ ϕ, and if H is a f.g. group such that H ⊨ ϕ, then G≅H. The first result is that all Baumslag–Solitar groups of the form 〈a, d | d-1ad = am〉 are QFA.
A f.g. group G is a prime model if and only if there is a tuple g1, … , gn generating G whose orbit (under the automorphisms of G) is definable by a first-order formula. The second result is that there are continuum many non-isomorphic f.g. groups that are prime models. In particular, not all are QFA.
Received: 2005-11-15
Revised: 2006-06-26
Published Online: 2007-06-18
Published in Print: 2007-05-23
© Walter de Gruyter
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Articles in the same Issue
- Transitivity properties for group actions on buildings
- Fields, values and character extensions in finite groups
- Primitive sharp permutation groups with large solvable subgroups
- On the size of the nilpotent residual in finite groups
- On the commuting complex of finite metanilpotent groups
- Soluble normally constrained pro-p-groups
- Comparing quasi-finitely axiomatizable and prime groups
- A finitely presented torsion-free simple group
- Andrews–Curtis groups and the Andrews–Curtis conjecture
- Residual finiteness of outer automorphism groups of certain tree products
- Commutator subgroups of Hantzsche–Wendt groups
