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**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.

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.

Reviewer: M.Deaconescu (Safat)

### MSC:

20E34 | General structure theorems for groups |

20F28 | Automorphism groups of groups |

20F16 | Solvable groups, supersolvable groups |

20E36 | Automorphisms of infinite groups |

### Keywords:

automizer; finite \(MD\)-groups; infinite \(MD\)-groups; infinite metabelian groups; torsion-free locally cyclic group; infinite dihedral group
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\textit{J. C. Lennox} and \textit{J. Wiegold}, Rend. Semin. Mat. Univ. Padova 89, 83--86 (1993; Zbl 0802.20026)

### 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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