Solvability of the radical in pseudo-finite groups with the DCC on centralizers up to finite index

TL;DR

This paper proves that in certain pseudo-finite groups with a chain condition on centralizers, the subgroup generated by solvable normal subgroups is itself solvable, and such groups cannot be finitely generated.

Contribution

It establishes the solvability of the radical in pseudo-finite groups under a chain condition on centralizers and rules out finitely generated examples with this property.

Findings

The subgroup generated by all solvable normal subgroups is solvable.

No finitely generated pseudo-finite group satisfies the chain condition on centralizers.

The results connect group structure with chain conditions on centralizers.

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.