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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
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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.

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

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