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?

Caranti, A., DALLA VOLTA, F. (2017). The multiple holomorph of a finitely generated abelian group. JOURNAL OF ALGEBRA, 481(1), 327-347 [10.1016/j.jalgebra.2017.03.006].

The multiple holomorph of a finitely generated abelian group

DALLA VOLTA, FRANCESCA
Ultimo
2017

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?
Articolo in rivista - Articolo scientifico
Commutative rings; Finitely-generated abelian groups; Holomorph; Multiple holomorph
English
22-mar-2017
2017
481
1
327
347
reserved
Caranti, A., DALLA VOLTA, F. (2017). The multiple holomorph of a finitely generated abelian group. JOURNAL OF ALGEBRA, 481(1), 327-347 [10.1016/j.jalgebra.2017.03.006].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/155146
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