On a question of Deaconescu about automorphisms. III
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.
scientific article; zbMATH DE number 1183132