The recovery of some abelian groups from their socles

Hill, Paul

doi:10.1090/S0002-9939-1982-0674080-1

The recovery of some abelian groups from their socles
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by Paul Hill

Proc. Amer. Math. Soc. 86 (1982), 553-560

DOI: https://doi.org/10.1090/S0002-9939-1982-0674080-1

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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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