Almost coproducts of finite cyclic groups

Paul Hill

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 4, page 795-804
  • ISSN: 0010-2628

Abstract

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A new class of p -primary abelian groups that are Hausdorff in the p -adic topology and that generalize direct sums of cyclic groups are studied. We call this new class of groups almost coproducts of cyclic groups. These groups are defined in terms of a modified axiom 3 system, and it is observed that such groups appear naturally. For example, V ( G ) / G is almost a coproduct of finite cyclic groups whenever G is a Hausdorff p -primary group and V ( G ) is the group of normalized units of the modular group algebra over Z / p Z . Several results are obtained concerning almost coproducts of cyclic groups including conditions on an ascending chain that implies that the union of the chain is almost a coproduct of cyclic groups.

How to cite

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Hill, Paul. "Almost coproducts of finite cyclic groups." Commentationes Mathematicae Universitatis Carolinae 36.4 (1995): 795-804. <http://eudml.org/doc/247724>.

@article{Hill1995,
abstract = {A new class of $p$-primary abelian groups that are Hausdorff in the $p$-adic topology and that generalize direct sums of cyclic groups are studied. We call this new class of groups almost coproducts of cyclic groups. These groups are defined in terms of a modified axiom 3 system, and it is observed that such groups appear naturally. For example, $V(G)/G$ is almost a coproduct of finite cyclic groups whenever $G$ is a Hausdorff $p$-primary group and $V(G)$ is the group of normalized units of the modular group algebra over $Z/pZ$. Several results are obtained concerning almost coproducts of cyclic groups including conditions on an ascending chain that implies that the union of the chain is almost a coproduct of cyclic groups.},
author = {Hill, Paul},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {primary groups; coproduct of cyclic groups; almost coproducts; third axiom of countability; -primary Abelian groups; -adic topology; weak axiom 3 systems of closed subgroups; ascending chain of subgroups; Hausdorff -groups; almost coproducts of cyclic groups; summands},
language = {eng},
number = {4},
pages = {795-804},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Almost coproducts of finite cyclic groups},
url = {http://eudml.org/doc/247724},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Hill, Paul
TI - Almost coproducts of finite cyclic groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 4
SP - 795
EP - 804
AB - A new class of $p$-primary abelian groups that are Hausdorff in the $p$-adic topology and that generalize direct sums of cyclic groups are studied. We call this new class of groups almost coproducts of cyclic groups. These groups are defined in terms of a modified axiom 3 system, and it is observed that such groups appear naturally. For example, $V(G)/G$ is almost a coproduct of finite cyclic groups whenever $G$ is a Hausdorff $p$-primary group and $V(G)$ is the group of normalized units of the modular group algebra over $Z/pZ$. Several results are obtained concerning almost coproducts of cyclic groups including conditions on an ascending chain that implies that the union of the chain is almost a coproduct of cyclic groups.
LA - eng
KW - primary groups; coproduct of cyclic groups; almost coproducts; third axiom of countability; -primary Abelian groups; -adic topology; weak axiom 3 systems of closed subgroups; ascending chain of subgroups; Hausdorff -groups; almost coproducts of cyclic groups; summands
UR - http://eudml.org/doc/247724
ER -

References

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  1. Fuchs L., Infinite Abelian Groups, vol. 2, Academic Press, New York, 1973. Zbl0338.20063MR0349869
  2. Hill P., On the classification of abelian groups, photocopied manuscript, 1967. 
  3. Hill P., Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups, Pacific Jour. Math. 42 (1972), 63-67. (1972) Zbl0251.20057MR0315018
  4. Hill P., Units of commutative modular group algebras, Jour. Pure and Applied Algebra, to appear. Zbl0806.16033MR1282838
  5. Hill P., Megibben C., On the theory and classification of abelian p -groups, Math. Zeit. 190 (1985), 17-38. (1985) Zbl0535.20031MR0793345
  6. Hill P., Ullery W., A note on a theorem of May concerning commutative group algebras, Proc. Amer. Math. Soc. 110 (1990), 59-63. (1990) Zbl0704.20007MR1039530
  7. Hill P., Ullery W., Almost totally projective groups, preprint. Zbl0870.20035MR1388614
  8. Kulikov L., On the theory of abelian groups of arbitrary power (Russian), Mat. Sb. 16 (1945), 129-162. (1945) MR0018180

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