W.H. Mills has determined, for a finitely generated abelian group G, the regular subgroups N≅G of S(G), the group of permutations on the set G, which have the same holomorph as G, that is, such that NS(G)(N)=NS(G)(ρ(G)), where ρ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of NS(G)(ρ(G)) in terms of commutative ring structures on G. We are led to solve, for the case of a finitely generated abelian group G, the following problem: given an abelian group (G,+), what are the commutative ring structures (G,+,⋅) such that all automorphisms of G as a group are also automorphisms of G as a ring? © 2017 Elsevier Inc.
The multiple holomorph of a finitely generated abelian group / Caranti, Andrea; F., Dalla Volta. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - STAMPA. - 481:(2017), pp. 327-347. [10.1016/j.jalgebra.2017.03.006]
The multiple holomorph of a finitely generated abelian group
Caranti, Andrea;
2017-01-01
Abstract
W.H. Mills has determined, for a finitely generated abelian group G, the regular subgroups N≅G of S(G), the group of permutations on the set G, which have the same holomorph as G, that is, such that NS(G)(N)=NS(G)(ρ(G)), where ρ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of NS(G)(ρ(G)) in terms of commutative ring structures on G. We are led to solve, for the case of a finitely generated abelian group G, the following problem: given an abelian group (G,+), what are the commutative ring structures (G,+,⋅) such that all automorphisms of G as a group are also automorphisms of G as a ring? © 2017 Elsevier Inc.File | Dimensione | Formato | |
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