Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The equivalence of high subgroups

Author: Paul Hill
Journal: Proc. Amer. Math. Soc. 88 (1983), 207-211
MSC: Primary 20K10; Secondary 20K27
MathSciNet review: 695242
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Two subgroups of a group $ G$ are called equivalent if there is an automorphism of $ G$ that maps one of the subgroups onto the other. Suppose that $ G$ is a $ p$-primary abelian group and that $ \lambda $ is an ordinal. A subgroup $ H$ of $ G$ is $ {p^\lambda }$-high in $ G$ if $ H$ is maximal in $ G$ with respect to having zero intersection with $ {p^\lambda }G$. Under certain conditions on the quotient group $ G/{p^\lambda }G$ slightly weaker than total projectivity, it is shown, for a given $ \lambda $, that any two $ {p^\lambda }$-high subgroups of $ G$ are equivalent. In particular, if $ G/{p^\omega }G$ is $ {p^{\omega + 1}}$-projective, the $ {p^\omega }$-high subgroups of $ G$ are all equivalent.

References [Enhancements On Off] (What's this?)

  • [1] D. Cutler, Primary abelian groups having all high subgroups isomorphic, Proc. Amer. Math. Soc. 83 (1981), 467-470. MR 627671 (82k:20086)
  • [2] L. Fuchs, Infinite abelian groups, Vols. 1, 2, Academic Press, New York, 1970, 1973.
  • [3] P. Hill, Certain pure subgroups of primary groups, Topics in Abelian Groups, Scott, Foresman, Chicago, Ill., 1963. MR 0169908 (30:151)
  • [4] -, On the classification of abelian groups, photocopied manuscript, Houston, 1967.
  • [5] -, Automorphisms of countable primary abelian groups, Proc. Amer. Math. Soc. 25 (1971), 135-140. MR 0255674 (41:334)
  • [6] P. Hill and C. Megibben, On primary groups with countable basic subgroups, Trans. Amer. Math. Soc. 124 (1966), 49-59. MR 0199260 (33:7409)
  • [7] J. Irwin and E. Walker, On $ N$-high subgroups of abelian groups, Pacific J. Math. 11 (1961), 1363-1374. MR 0136653 (25:119a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20K10, 20K27

Retrieve articles in all journals with MSC: 20K10, 20K27

Additional Information

Keywords: Abelian $ p$-groups, high subgroup, equivalent subgroups, $ {p^\alpha }$-projective, extending isomorphisms, automorphism
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society