In pseudo-finite groups satisfying DCC on centralizers up to finite index, the solvable radical is solvable, and no finitely generated such group exists whose definable sections satisfy the condition.
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Pseudo-finite groups are infinite groups that satisfy exactly the same first-order logical sentences as some finite groups. The descending chain condition on centralizers up to finite index is a finiteness-like restriction: any descending sequence of centralizers must stabilize once subgroups of finite index are ignored. The main result states that the solvable radical, defined as the subgroup generated by all solvable normal subgroups, must itself be solvable under these assumptions. A second result rules out the existence of any finitely generated pseudo-finite group in which all definable sections obey the same chain condition. These statements are proved using tools from model theory and infinite group theory.
Core claim
The subgroup generated by all solvable normal subgroups in a pseudo-finite group with the descending chain condition on centralizers up to finite index is solvable.
Load-bearing premise
The group is pseudo-finite and satisfies the descending chain condition on centralizers up to finite index; the argument relies on model-theoretic properties of definable sets and sections.
read the original abstract
The subgroup generated by all solvable normal subgroups in a pseudo-finite group with the descending chain condition on centralizers up to finite index is solvable. Additionally, there is no finitely generated pseudo-finite group whose definable sections satisfy such a chain condition on centralizers.
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The result rests on the standard definitions of pseudo-finite groups and the DCC condition, which are drawn from prior literature in model theory. No free parameters or new entities are introduced in the abstract.
axioms (1)
standard mathAxioms of first-order logic and the definition of groups The paper works inside ordinary model theory and group theory.
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