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On a question of Deaconescu about automorphisms

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On a question of Deaconescu about automorphisms RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA JOHNC. LENNOX JAMESWIEGOLD On a question of Deaconescu about automorphisms Rendiconti del Seminario Matematico della Università di Padova, tome 89 (1993), p. 83-86. <http://www.numdam.org/item?id=RSMUP_1993__89__83_0> © Rendiconti del Seminario Matematico della Università di Padova, 1993, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est consti- tutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On a Question of Deaconescu About Automorphisms. JOHN C. LENNOX - JAMES WIEGOLD 1. Introduction. At a recent meeting in Barnaul, Siberia, Marian Deaconescu posed the following problem. 1.1. Do there exist infinite groups G such that Aut H = = NG (H ) /CG (H ) for all subgroups H of G? Let us call (finite or infinite) groups with this property MD-groups. We note at the outset that the infinite dihedral group D has the MD- property, and this is fairly easy to prove. Theorem 2.1 characterizes the infinite metabelian MD-groups; there are just 2xo of them, and they all have very simple structure. A corollary is that D is the only finitely generated infinite metabelian MD-group. It is quite probable that the groups covered in Theorem 2.1 are the only infinite MD-groups, because automorphism groups tend to be in- soluble and big. For example, every free group Fm of infinite rank m has F2m in its automorphism group; thus, by the splitting property of free groups, any MD-group containing a subgroup whose automor- phism gro

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