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The recovery of some abelian groups from their socles


Author: Paul Hill
Journal: Proc. Amer. Math. Soc. 86 (1982), 553-560
MSC: Primary 20K25; Secondary 20K10, 20K27
DOI: https://doi.org/10.1090/S0002-9939-1982-0674080-1
MathSciNet review: 674080
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Abstract: The first main result of this paper is that summable isotype subgroups inherit total projectivity. This means: ($ \ast $) if an isotype subgroup of a totally projective group has a free socle (viewed as a valuated vector space) then the socle completely determines the subgroup up to isomorphism. The next major result is that ($ \ast $) does not generalize to the case where the socle is that of a totally projective group of length exceeding $ \Omega $, nor does ($ \ast $) generalize to the case where the socle is that of an $ S$-group of length $ \Omega $. Finally, it is shown that if, in addition to $ H$ being isotype in a d.s.c. group $ G$, it is also known that $ K/H$ is divisible where $ K$ is the closure of $ H$ in $ G$ relative to the $ {p^\Omega }$-topology, then ($ \ast $) again prevails when the socle is that of an $ S$-group.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0674080-1
Keywords: Primary abelian group, isotype, summable, totally projective, separable subgroup, d.s.c., $ S$-group, socle, valuated vector space
Article copyright: © Copyright 1982 American Mathematical Society

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