Existentiell abgeschlossene Sylowturmgruppen. (Existentially closed Sylow tower groups)
type: Item
instance of: scholarly article
title: Existentiell abgeschlossene Sylowturmgruppen. (Existentially closed Sylow tower groups)
zbMATH Open document ID: 0642.20036
DOI: 10.1007/BF01198621
author: Heiner Ensel
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: For a set \(\pi\) of primes we denote by \(ST_{\pi}\) the class of all locally finite \(\pi\)-groups in which there exists a normal series \(\{(S_ p,T_ p)|\) \(p\in \pi \}\) (of an order type \(<\}\) such that the factor group \(S_ p/T_ p\) is a p-group for every \(p\in \pi\). We prove some properties of existentially closed \(ST_{\pi}\)-groups G, in which the subgroups \(S_ p\) and \(T_ p\) possess complements \(S^ *_ p\) and \(T^ *_ p\) in G. The main results are the following: The group \(T^ *_ p\) is an existentially closed \(ST_{\pi _ 1}\)-group, where \(\pi _ 1=\{q\in \pi |\) \(p\leq q\}\), and \(S^ *_ p\) is an existentially closed \(ST_{\pi _ 1\setminus p}\)-group. The groups G, \(S_ p\) and \(T_ p\) are verbally complete. Furthermore the group G possesses a unique chief series. The order type of this series is a dense, linear order without endpoints. If p is a non-maximal prime in \(\pi\), relative to the order \(<\), then each chief factor M/N in G with \(T_ p\subseteq N\subseteq M\subseteq S_ p\) is an infinite, non-central, elementary-abelian p- group. Some of these results we can prove without the additional assumption, that the groups \(S_ p\) and \(T_ p\) possess complements in G.
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 4046019
zbMATH Keywords: locally finite \(\pi \) -groups
zbMATH Keywords: normal series
zbMATH Keywords: existentially closed \(ST_{\pi }\)-groups
zbMATH Keywords: complements
zbMATH Keywords: verbally complete
zbMATH Keywords: chief series
MaRDI profile type: Publication
cites work: Sylowturmgruppen
cites work: Some Constructions for Locally Finite Groups
cites work: Forcing, arithmetic, division rings
cites work: Q5729345
cites work: Existentially closed L\({\mathfrak X}\)-groups
cites work: Existentially closed groups in locally finite group classes
cites work: Existentially closed locally finite p-groups
cites work: Existenziell abgeschlossene lokal endliche p-Gruppen
cites work: On amalgams of periodic groups
cites work: On the structure of standard wreath products of groups
description: scientific article; zbMATH DE number 4046019