ON THE HOPFICITY OF COMPLETELY DECOMPOSABLE TORSION-FREE ABELIAN GROUPS

E. Kaygorodov, M. Chanchieva

IJPAM: Volume 113, No. 2 (2017)

Title

ON THE HOPFICITY OF COMPLETELY DECOMPOSABLE
TORSION-FREE ABELIAN GROUPS

Authors

Evgeniy V. Kaygorodov

, Marina V. Chanchieva

$^{1,2}$ Faculty of Physics and Mathematics
Department of Mathematics and Informatics
Gorno-Altaisk State University
649000, GASU, Lenkin Str. 1, Gorno-Altaisk, Altai Republic, RUSSIA

Abstract

The property of Hopficity is studied in a well-known class of Abelian groups - completely decomposable torsion-free groups. Examples of non-Hopfian completely decomposable torsion-free group are constructed.

History

Received: December 4, 2016
Revised: March 1, 2017
Published: March 19, 2017

AMS Classification, Key Words

AMS Subject Classification: 16Txx, 20Kxx
Key Words and Phrases: Hopficity, Abelian group, completely decomposable groups, homogeneous group, torsion-free group

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1: A.I. Malcev, On isomorphic matrix representations of infinite groups, Recreational Mathematics, 8 (1940), 405-422.
2: B.H. Neumann, A two-generator group isomorphic to a proper factor group, Journal of the London Mathematical Society, 25 (1950), 247-248.
3: G. Higman, A finitely related group with an isomorphic proper factor group, Journal of the London Mathematical Society, 26 (1951), 59-61.
4: G. Baumslag, D. Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society, 68 (1962), 199-201.
5: D. Meier, Non-Hopfian groups, Journal of the London Mathematical Society, 26 (1982), 265-270.
6: Y. de Cornulier, Finitely presentable, non-Hopfian groups with Kazhdan's Property (T) and infinite outer automorphism group, Proceedings of the American Mathematical Society, 135 (2007), 951-959.
7: V.H. Mikaelian, On finitely generated soluble non-Hopfian groups, Journal of Mathematical Sciences, 166 (2010), 743-755.
8: R. Hirshon, Some theorems on Hopficity, Transactions of the American Mathematical Society, 141 (1969), 229-244.
9: R. Hirshon, On Hopfian groups, Pacific Journal of Mathematics, 32 (1970),753-766.
10: R. Hirshon, A conjecture on Hopficity and related results, Archiv der Mathematik (Basel), 22 (1971), 449-455.
11: R. Hirshon, The center and the commutator subgroup in Hopfian groups, Arkiv för Matematik, 9 (1971), 181-192.
12: R. Hirshon, The direct product of Hopfian group with a group with cyclic center, Arkiv för Matematik, 10 (1972), 231-234.
13: R. Hirshon, The direct product of a Hopfian group with p-group, Archiv der Mathematik (Basel), 26 (1975), 470-479.
14: R. Hirshon, Some properties of endomorphisms in residually finite groups, Journal of the Australian Mathematical Society. Series A, 24 (1977), 117-120.
15: R. Hirshon, Misbehaved direct products, Expositiones Mathematicae, 20 (2002), 365-374.
16: G. Baumslag, Hopficity and Abelian groups, Topics in Abelian groups, Proceedings of the New Mexico Symposium on Abelian Groups, Scott-Foresman-Chicago, New Mexico State University (1962), 331-335.
17: G. Baumslag, On Abelian Hopfian groups. I, Mathematische Zeitschrift, 78 (1962), 53-54.
18: G. Baumslag, Products of Abelian Hopfian groups, Journal of the Australian Mathematical Society, 8 (1968), 322-326.
19: A.L.S. Corner, Three examples on Hopficity in torsion-free Abelian groups, Acta Mathematica Academiae Scientiarum Hungaricae, 16 (1965), 303-310.
20: J.M. Irwin, J. Takashi, A quasi-decomposable Abelian group without proper isomorphic quotient groups and proper isomorphic subgroups, Pacific Journal of Mathematics, 29 (1969), 151-160.
21: B. Goldsmith, K. Gong, On super and hereditarily Hopfian and co-Hopfian Abelian groups, Archiv der Mathematik, 99 (2012), 1-8.
22: B. Goldsmith, K. Gong, On adjoint entropy of Abelian groups, Communications in Algebra, 40 (2012), 972-987.
23: B. Goldsmith, K. Gong, A note on Hopfian and co-Hopfian Abelian groups, AMS Forthcoming, Dublin (2012), 1-9.
24: B. Goldsmith, K. Gong, On some generalizations of Hopfian and co-Hopfian Abelian groups, Acta Mathematica Hungarica, 139 (2013), 393-398.
25: G. Braun, L. Strüngmann, The independence of the notions of Hopfian and co-Hopfian Abelian p-groups, Proceedings of the American Mathematical Society, 143 (2015), 3331-3341.
26: E.V. Kaigorodov, Hopfian algebraically compact Abelian groups, Algebra and Logic, 52 (2014), 442-447.
27: L. Fuchs, Infinite Abelian Groups, Vol. I, Academic Press, New York (1970).
28: L. Fuchs, Infinite Abelian Groups, Vol. II, Academic Press, New York (1973).
29: P.A. Krylov, A.V. Mikhalev, A.A. Tuganbaev, Endomorphism Rings of Abelian Groups, Kluwer Academic Publishers, Dordrecht, Boston, London (2003).
30: A.G. Kurosh, Lectures on General Algebra, Chelsea Publishing Co., New York (1963).
31: D.M. Arnold, Finite rank torsion free Abelian groups and rings, Lecture Notes in Mathematics, Vol. 931, Springer-Verlag, Berlin, Heidelberg, New York (1982).

How to Cite?

DOI: 10.12732/ijpam.v113i2.11 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 113
Issue: 2
Pages: 327 - 334

Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).