On a question of Deaconescu about automorphisms. (English) Zbl 0802.20026

Let \(G\) be a finite group and let \(H\) be a subgroup of \(G\). The automizer of \(H\) in \(G\) is the group \(N_ G(H)/C_ G(H) = A(H)\) and it is known that \(A(H)\) is embedded into \(\text{Aut}(H)\). A group \(G\) is called by the authors an \(MD\)-group if \(A(H) \cong \text{Aut}(H)\) for every \(H \leq G\). The reviewer proved [Am. Math. Mon. 99, No. 10 (1992), Problem 10270] that the groups \(S_ n\), \(n \leq 3\) are the only finite \(MD\)-groups and asked what can be said about the infinite \(MD\)-groups.
In the paper under review, the authors determine the structure of the infinite metabelian groups. They prove that an infinite metabelian group \(G\) is an \(MD\)-group iff \(G = \langle A,x\rangle\), where \(A\) is a nontrivial torsion-free locally cyclic group and \(x\) is an involution such that: i) \(a^ x = a^{-1}\) for all \(a \in A\), ii) the type of \(A\) is finite at every prime. – As direct consequences of this result, it is shown that: (a) There exist precisely \(2^{\aleph_ 0}\) infinite metabelian \(MD\)-groups and b) The infinite dihedral group is the only infinite finitely generated metabelian \(MD\)-group. The authors also give a heuristic argument to support the conjecture that every infinite \(MD\)- group must be metabelian.


20E34 General structure theorems for groups
20F28 Automorphism groups of groups
20F16 Solvable groups, supersolvable groups
20E36 Automorphisms of infinite groups
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