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FIRST-ORDER RECOGNIZABILITY IN FINITE AND PSEUDOFINITE GROUPS

Part of: Representation theory of groups Model theory Foundations

Published online by Cambridge University Press:  20 July 2020

YVES CORNULIER  and
JOHN S. WILSON
YVES CORNULIER
Affiliation:
CNRS AND UNIV LYON UNIV CLAUDE BERNARD LYON 1 INSTITUT CAMILLE JORDAN 43 BLVD DU 11 NOVEMBRE 1918 69622VILLEURBANNE, FRANCEE-mail: cornulier@math.univ-lyon1.fr
JOHN S. WILSON
Affiliation:
CHRIST’S COLLEGECAMBRIDGE CB2 3BU, UK and MATHEMATISCHES INSTITUT UNIVERSITÄT LEIPZIGLEIPZIG, GERMANYE-mail: jsw13@cam.ac.uk
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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

pseudofinite groupspseudo-soluble groupspseudo-nilpotent groupsfirst-order theoryFrattini subgroup

MSC classification

Primary: 03C60: Model-theoretic algebra
Secondary: 20A15: Applications of logic to group theory 03C13: Finite structures 03C20: Ultraproducts and related constructions 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20D15: Nilpotent groups, $p$-groups 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 852 - 867
Copyright
© The Association for Symbolic Logic 2020

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References

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