Another summable $C_{\Omega }$-group
Author:
Doyle O. Cutler
Journal:
Proc. Amer. Math. Soc. 26 (1970), 43-44
MSC:
Primary 20.30
DOI:
https://doi.org/10.1090/S0002-9939-1970-0262355-1
MathSciNet review:
0262355
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Abstract: An example is given of a $p$-primary Abelian group $G$ having the following properties: $G$ is summable; the length of $G$ is $\Omega$; the $\alpha$th Ulm invariant of $G$ is one for all $\alpha < \Omega$; if $\alpha < \Omega$, any ${p^\alpha }G$-high subgroup of $G$ is countable; $G/{p^\alpha }G$ is countable for all $\alpha < \Omega$; and $G$ is not ${p^\beta }$-projective for any ordinal $\beta$.
- Doyle O. Cutler, On the structure of primary abelian groups of countable Ulm type, Trans. Amer. Math. Soc. 152 (1970), 503–518. MR 276330, DOI https://doi.org/10.1090/S0002-9947-1970-0276330-9
- Paul Hill, A summable $C_{\Omega }$-group, Proc. Amer. Math. Soc. 23 (1969), 428–430. MR 245674, DOI https://doi.org/10.1090/S0002-9939-1969-0245674-6
- Paul Hill and Charles Megibben, On direct sums of countable groups and generalizations, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 183–206. MR 0242943
- Kin-ya Honda, Realism in the theory of abelian groups. III, Comment. Math. Univ. St. Paul. 12 (1964), 75–111. MR 162848 Charles Megibben, A generalization of the classical theory of primary groups, Notices Amer. Math. Soc. 15 (1968), 1025. Abstract #661-5.
- R. J. Nunke, Purity and subfunctors of the identity, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman and Co., Chicago, Ill., 1963, pp. 121–171. MR 0169913
- R. J. Nunke, Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182–212. MR 218452, DOI https://doi.org/10.1007/BF01135839
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Additional Information
Keywords:
Summable,
<IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$p$">-primary Abelian group,
length <IMG WIDTH="20" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\Omega$">,
not <!– MATH ${p^\beta }$ –> <IMG WIDTH="27" HEIGHT="45" ALIGN="MIDDLE" BORDER="0" SRC="images/img4.gif" ALT="${p^\beta }$">-projective
Article copyright:
© Copyright 1970
American Mathematical Society