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Subgroup transitivity in abelian groups

Authors: Paul Hill and Jane Kirchner West
Journal: Proc. Amer. Math. Soc. 126 (1998), 1293-1303
MSC (1991): Primary 20K10, 20K27; Secondary 20K30, 20E36
MathSciNet review: 1443830
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Abstract: We generalize an appropriate modification of the classical notion of transitivity in abelian $p$-groups from one that is based on elements to one based on subgroups. We consider those $p$-groups that are transitive in the sense that there is an automorphism of the group that maps one isotype subgroup $H$ onto any other isotype subgroup $H'$, unless this is impossible due to the simple reason that either the subgroups are not isomorphic or the quotient groups are not (as valuated groups when endowed with the coset valuation). Slight variations of this are used to define the classes of strongly transitive and strongly U-transitive groups. The latter class is studied in some detail in this paper, and it is shown that every $C_\Omega$-group is strongly transitive with respect to countable isotype subgroups.

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

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Additional Information

Paul Hill
Affiliation: Department of Mathematics, Auburn University, Alabama 36849

Jane Kirchner West
Affiliation: Department of Natural Sciences, Colby-Sawyer College, New London, New Hampshire 03257

Keywords: Primary groups, subgroup transitivity, equivalent subgroups, automorphisms, totally projective groups
Received by editor(s): May 12, 1996
Received by editor(s) in revised form: October 23, 1996
Additional Notes: The first author was supported by NSF grant DMS 92-08199
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1998 American Mathematical Society

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