Duality in some classes of torsion-free Abelian groups of finite rank.

*(English. Russian original)*Zbl 0633.20031
Sib. Math. J. 27, 563-571 (1986); translation from Sib. Mat. Zh. 27, No. 4(158), 117-127 (1986).

Let \(\sigma\), \(\tau\) be a pair of types of torsion-free (abelian) groups of rank 1 which are determined by characteristics \((k_ p)\), \((m_ p)\) such that \(k_ p\leq m_ p\) for all primes \(p\). A torsion-free group \(A\) of finite rank \(n\) belongs to the class \(D^{\tau}_{\sigma}\) iff there exists a free subgroup \(J\) of rank \(n\) of \(A\) such that any \(p\)-primary component of \(A/J\) is a direct sum of \(n\) groups isomorphic either to \(Z(p^{k_ p})\) or to \(Z(p^{m_ p})\). For primes \(p\) such that \(k_ p<m_ p\) let \(s_ p(A)\), \(t_ p(A)\) denote the number of isomorphic summands so \((s_ p(A),t_ p(A))\) are determined up to an equivalence (similar to that of characteristics). It is easy to see that the type of any element of any group in \(D^{\tau}_{\sigma}\) is greater than or equal to \(\sigma\). A detailed study of groups \(A/p^{m_ p}A\) and of \(p\)-adic completions of \(A\) leads to a notion of duality in the class \(D^{\tau}_{\sigma}\), i.e. a contravariant functor \(-^*: D^{\tau}_{\sigma}\to D^{\tau}_{\sigma}\) (where \(D^{\tau}_{\sigma}\) is considered as the category of groups A with a fixed basis of \(J\) and with quasihomomorphisms as maps) such that 1) rank \(A^*=rank A\), 2) \(s_ p(A^*)=t_ p(A)\), \(t_ p(A^*)=s_ p(A)\) for almost all \(p\), 3) \(A^{**}=A\), \(\phi^{**}=\phi\) for all groups \(A\) and all quasihomomorphisms \(\phi\). Any two such dualities are equivalent. This duality is compared with those defined by Warfield and Arnold; e.g. it is shown that if \(m_ p=\infty\) implies \(k_ p=\infty\) then \(-^*=\operatorname{Hom}(-,R)\) where \(R\) is a torsion free group of rank 1 and of type \(\sigma+\tau\).

Reviewer: St.Balcerzyk

##### MSC:

20K15 | Torsion-free groups, finite rank |

20K40 | Homological and categorical methods for abelian groups |

##### Keywords:

types; free subgroup; \(p\)-primary component; summands; \(p\)-adic completions; duality; contravariant functor; category of groups; quasihomomorphisms; torsion free group
PDF
BibTeX
XML
Cite

\textit{A. A. Fomin}, Sib. Math. J. 27, 563--571 (1986; Zbl 0633.20031); translation from Sib. Mat. Zh. 27, No. 4(158), 117--127 (1986)

Full Text:
DOI

##### References:

[1] | L. Ya. Kulikov, ?Generalized primary groups,? Tr. Mosk. Mat. Obshch.,1, 247-326 (1952). |

[2] | A. A. Beaumont and R. S. Pierce, ?Torsion-free rings,? Illinois J. Math.,5, No. 1, 61-98 (1961). · Zbl 0108.03802 |

[3] | R. Baer, ?Abelian groups without elements of finite order,? Duke Math. J.,3, No. 1, 68-122 (1937). · Zbl 0016.20303 |

[4] | D. M. Arnold, ?A duality for torsion-free modules of finite rank over a discrete valuation ring,? Proc. London Math. Soc.,24, No. 3, 204-216 (1972). · Zbl 0237.13016 |

[5] | D. M. Arnold, ?A duality for quotient divisible Abelian groups of finite rank,? Pac. J. Math.,42, No. 1, 11-15 (1972). · Zbl 0262.20062 |

[6] | R. B. Warfield, ?Homomorphisms and duality for torsion-free groups,? Math. Z.,107, No. 1, 189-200 (1968). · Zbl 0169.03602 |

[7] | L. Fuchs, Infinite Abelian Groups, Vol. 1, Academic Press, New York-London (1970). · Zbl 0209.05503 |

[8] | F. Richman, ?A class of rank 2 torsion-free groups,? in: Studies on Abelian Groups, Paris (1968), pp. 327-333. |

[9] | C. E. Murley, ?The classification of certain classes of torsion-free Abelian groups,? Pac. J. Math.,40, No. 3, 647-665 (1972). · Zbl 0261.20045 |

[10] | L. Ya. Kulikov, Algebra and Number Theory [in Russian], Vysshaya Shkola, Moscow (1979). |

[11] | M. Bourbaki, Algebra I, Hermann, Paris (1974), Chaps. I?III. |

[12] | A. A. Fomin, ?Tensor products of torsion-free Abelian groups,? Sib. Mat. Zh.,16, No. 5, 1071-1080 (1975). |

[13] | A. A. Fomin, ?Monogeneous ?-groups,? in: Proceedings 7th All-Union Symposium on Group Theory, Krasnoyarsk State Univ., 128 (1980). |

[14] | D. W. Dubois, ?Cohesive groups and p-adic integers,? Publ. Math. Debrecen,12, No. 1, 51-58 (1965). · Zbl 0136.29002 |

[15] | A. A. Fomin, ?Abelian groups with free subgroups of infinite index and their endomorphism rings,? mat. Zametki,36, No. 2, 179-187 (1984). |

[16] | A. A. Fomin, ?Abelian groups with free pure subgroups,? in: Proceedings 9th All-Union Symposium on Group Theory, MGPI im. V. I. Lenina, Moscow (1984), p. 162. · Zbl 0571.20047 |

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.