## 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.

### MSC:

 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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### References:

 [1] L. Fuchs , Abelian Groups , Hungarian Academy of Sciences , Budapest ( 1958 ). MR 106942 | Zbl 0091.02704 · Zbl 0091.02704 [2] W. Gaschütz , Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen , J. Algebra , 4 ( 1966 ), pp. 1 - 2 . MR 193144 | Zbl 0142.26001 · Zbl 0142.26001
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