Existentially closed L\({\mathfrak X}\)-groups
type: Item
instance of: scholarly article
title: Existentially closed L\({\mathfrak X}\)-groups
zbMATH Open document ID: 0598.20036
author: Felix Leinen
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1986-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108020
review text: If \({\mathfrak Y}\) is a class of groups, then a \({\mathfrak Y}\)-group G is existentially closed if every system of finitely many equations and inequations with coefficients from G, which has a solution in some \({\mathfrak Y}\)-supergroup of G, can already be solved in G. Properties of such groups when \({\mathfrak Y}\) is the class of all locally finite groups follow from the work of \textit{P. Hall} [J. Lond. Math. Soc. 34, 305-319 (1959; Zbl 0088.023)] and when \({\mathfrak Y}\) is the class of all locally finite p-groups from the work of \textit{B. Maier} [Arch. Math. 37, 113-128 (1981; Zbl 0474.20015)]. In this paper, the author considers classes \({\mathfrak Y}\) of the form L\({\mathfrak X}\) (the class of all locally \({\mathfrak X}\)-groups) when \({\mathfrak X}\) is closed under the taking of subgroups, extensions, quotient groups and cartesian powers of finitely generated subgroups. (Not all these properties are required in every case however.) A group is said to be verbally complete if every element of the group is an instance of every non-trivial word (taken from some fixed countable free group). It is shown that e.c. L\({\mathfrak X}\)-groups are verbally complete in many cases; for example when the group is periodic or when the class \({\mathfrak X}\) contains a non-trivial p-group for each prime p. The general question, however, remains open. More particularly, if \({\mathfrak X}\) is closed under taking subgroups, extensions, quotient groups and cartesian powers of finitely generated subgroups, is an e.c. L\({\mathfrak X}\)-group verbally complete ? The next section considers characteristic subgroups of e.c. L\({\mathfrak X}\)- groups. Again, it is shown that in many cases these must be proper or trivial but the general case is not fully understood. In particular, the author asks under what conditions an e.c. L\({\mathfrak X}\)-group (\({\mathfrak X}\) locally finite) is characteristically simple. The normal subgroup structure of countable e.c. …
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E06
zbMATH DE Number: 3964169
zbMATH Keywords: existentially closed groups
zbMATH Keywords: system of equations
zbMATH Keywords: verbally complete groups
zbMATH Keywords: normal subgroups totally ordered by inclusion
zbMATH Keywords: locally finite groups
zbMATH Keywords: locally finite p-groups
zbMATH Keywords: finitely generated subgroups
zbMATH Keywords: characteristic subgroups
zbMATH Keywords: characteristically simple
zbMATH Keywords: normal subgroup structure
zbMATH Keywords: countable e.c. L\({\mathfrak X}\)-groups
zbMATH Keywords: unique chief series
zbMATH Keywords: automorphisms
zbMATH Keywords: isomorphism types
reviewed by: J. R. J. Groves
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3964169