Submission date: 5 November 2019

Abstract:

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with 'solubility' replaced by 'nilpotence' and 'perfectness', among others, are false.
These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini's theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

Mathematics Subject Classification: Primary 03C60, 20A15, Secondary 03C13, 03C20, 20D10, 20D15, 20D20

Keywords and phrases:

Full text arXiv 1911.01976: pdf, ps.