×

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
PDF BibTeX XML Cite
Full Text: Numdam EuDML

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.