First-order recognisability in finite and pseudofinite groups
Group Theory
2021-05-11 v2 Logic
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.
Keywords
Cite
@article{arxiv.1911.01976,
title = {First-order recognisability in finite and pseudofinite groups},
author = {Yves Cornulier and John S. Wilson},
journal= {arXiv preprint arXiv:1911.01976},
year = {2021}
}
Comments
18 pages (v2: Proposition 5.1 added)