In this paper, we study holomorphically isomorphic Abelian groups, i.e., Abelian groups with isomorphic holomorphs. We also study a generalization of the concept of holomorphic isomorphism, namely, almost holomorphic isomorphism, which is deeply connected with normal Abelian subgroups of holomorphs of Abelian groups. Torsion-free Abelian groups that are determined by their holomorphs are highlighted from different classes. In particular, it has been found that any homogeneous separable group can be determined by its holomorph in the class of all Abelian groups.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
I. Kh. Bekker, “On holomorphs of Abelian groups,” Sib. Mat. Zh., 5, No. 6, 1228–1238 (1964).
I. Kh. Bekker, “On holomorphs of non-reduced Abelian groups,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 3–8 (1968).
I. Kh. Bekker, “On holomorphs of torsion-free Abelian groups,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 3–13 (1974).
I. Kh. Bekker, “Abelian groups with isomorphic holomorphs,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 97–99 (1975).
I. Kh. Bekker, “Abelian holomorphic groups,” in: Int. Conf. “All-Siberian Readings on Mathematics and Mechanics.” Selected Reports. V. 1. Mathematics [in Russian] (1997), pp. 43–47.
I. Kh. Bekker and S. Ya. Grinshpon, “Almost holomorphically isomorphic primary Abelian groups,” in: Groups and Modules. Interacademic Subject Collection of Scientific Works (1976), pp. 90–103.
L. Fuchs, Infinite Abelian Groups, V. 2 [Russian translation], Mir, Moscow (1977).
I. E. Grinshpon, “Normal subgroups of holomorphs of Abelian groups and almost holomorphic isomorphism,” Fundam. Prikl. Mat., 13, No. 3, 9–16 (2007).
S. Ya. Grinshpon, “Almost holomorphically isomorphic Abelian groups,” Tr. Tomsk. Univ., 220, Problems of Mathematics, No. 3, 78–84 (1975).
S. Ya. Grinshpon, “On the structure of fully invariant subgroups of torsion-free Abelian groups,” in: Abelian Groups and Modules, Tomsk (1982), pp. 56–92.
S. Ya. Grinshpon, “Fully invariant subgroups of Abelian groups and full transitivity,” Fundam. Prikl. Mat., 8, No. 2, 407–473 (2002).
G. A. Miller, “On the multiple holomorph of a group,” Math. Ann., 66, 133–142 (1908).
W. H. Mills, “Multiple holomorphs of finitely generated Abelian groups,” Trans. Am. Math. Soc., 71, No. 3, 379–392 (1950).
W. H. Mills, “On the non-isomorphism of certain holomorphs,” Trans. Am. Math. Soc., 74, No. 3, 428–443 (1953).
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 8, pp. 35–46, 2011/12.
About this article
Cite this article
Grinshpon, S.Y., Grinshpon, I.E. Determinateness of Torsion-Free Abelian Groups by Their Holomorphs and Almost Holomorphic Isomorphism. J Math Sci 197, 605–613 (2014). https://doi.org/10.1007/s10958-014-1742-1
- Abelian Group
- Normal Subgroup
- Nonzero Element
- Homogeneous Group
- Direct Summand