On a question of Deaconescu about automorphisms. III
type: Item
instance of: scholarly article
title: On a question of Deaconescu about automorphisms. III
zbMATH Open document ID: 0908.20020
author: Viatcheslav N. Obraztsov
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1999-03-18T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108477
review text: The main result of this impressive paper is too technical to be stated here; it ensures that certain free amalgams of families of groups can be embedded into infinite simple groups with prescribed properties. The result is a modified version of a previous theorem of the author [Bull. Aust. Math. Soc. 54, No. 2, 221-240 (1996; Zbl 0864.20018)], so it is possible to construct examples of infinite simple MD-groups. An MD-group is a group \(G\) with the property that \(N_G(H)/C_G(H)\cong\Aut(H)\) for every subgroup \(H\) of \(G\). \textit{H. Smith} and \textit{J. Wiegold} [Part II, Rend. Semin. Mat. Univ. Padova 91, 61-64 (1994; Zbl 0807.20028)] proved that the infinite dihedral group is the only infinite MD-group which is finitely generated and locally soluble. The following two corollaries of the main result of this paper are significant from the point of view of infinite MD-groups: Corollary 1. For each infinite cardinal \(\alpha\) there exists a simple MD-group of cardinality \(\alpha\). Corollary 2. There exists a continuum of pairwise nonisomorphic 2-generator simple infinite MD-groups in which every maximal proper subgroup is infinite dihedral.
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 1183132
zbMATH Keywords: free amalgams
zbMATH Keywords: infinite simple groups
zbMATH Keywords: simple MD-groups
zbMATH Keywords: infinite dihedral group
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
cites work: On a question of Deaconescu about automorphisms
cites work: Q4735339
cites work: On infinite complete groups
cites work: Q4342757
cites work: Embedding into groups with well-described lattices of subgroups
cites work: Q4002122
cites work: Economical embeddings of countable groups
cites work: On a question of Deaconescu about automorphisms. II
description: scientific article; zbMATH DE number 1183132