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On a question of Deaconescu about automorphisms. III. (English) Zbl 0908.20020

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 \(N_G(H)/C_G(H)\cong\operatorname{Aut}(H)\) for every subgroup \(H\) of \(G\). H. Smith and 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.

MSC:

20E22 Extensions, wreath products, and other compositions of groups
20E32 Simple groups
20F28 Automorphism groups of groups
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
20E28 Maximal subgroups
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References:

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