The generalized criterion of Dieudonné for valuated p -groups

Peter Vassilev Danchev

Acta Mathematica Universitatis Ostraviensis (2006)

  • Volume: 14, Issue: 1, page 17-19
  • ISSN: 1804-1388

Abstract

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We prove that if G is an abelian p -group with a nice subgroup A so that G / A is a Σ -group, then G is a Σ -group if and only if A is a Σ -subgroup in G provided that A is equipped with a valuation induced by the restricted height function on G . In particular, if in addition A is pure in G , G is a Σ -group precisely when A is a Σ -group. This extends the classical Dieudonné criterion (Portugal. Math., 1952) as well as it supplies our recent results in (Arch. Math. Brno, 2005), (Bull. Math. Soc. Sc. Math. Roumanie, 2006) and (Acta Math. Sci., 2007).

How to cite

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Danchev, Peter Vassilev. "The generalized criterion of Dieudonné for valuated $p$-groups." Acta Mathematica Universitatis Ostraviensis 14.1 (2006): 17-19. <http://eudml.org/doc/35157>.

@article{Danchev2006,
abstract = {We prove that if $G$ is an abelian $p$-group with a nice subgroup $A$ so that $G/A$ is a $\Sigma $-group, then $G$ is a $\Sigma $-group if and only if $A$ is a $\Sigma $-subgroup in $G$ provided that $A$ is equipped with a valuation induced by the restricted height function on $G$. In particular, if in addition $A$ is pure in $G$, $G$ is a $\Sigma $-group precisely when $A$ is a $\Sigma $-group. This extends the classical Dieudonné criterion (Portugal. Math., 1952) as well as it supplies our recent results in (Arch. Math. Brno, 2005), (Bull. Math. Soc. Sc. Math. Roumanie, 2006) and (Acta Math. Sci., 2007).},
author = {Danchev, Peter Vassilev},
journal = {Acta Mathematica Universitatis Ostraviensis},
keywords = {height valuation; valuated subgroups; countable unions of subgroups; $\Sigma $-groups; valuated -groups; height valuations; valuated subgroups; nice subgroups; pure subgroups; Abelian -groups; -groups},
language = {eng},
number = {1},
pages = {17-19},
publisher = {University of Ostrava},
title = {The generalized criterion of Dieudonné for valuated $p$-groups},
url = {http://eudml.org/doc/35157},
volume = {14},
year = {2006},
}

TY - JOUR
AU - Danchev, Peter Vassilev
TI - The generalized criterion of Dieudonné for valuated $p$-groups
JO - Acta Mathematica Universitatis Ostraviensis
PY - 2006
PB - University of Ostrava
VL - 14
IS - 1
SP - 17
EP - 19
AB - We prove that if $G$ is an abelian $p$-group with a nice subgroup $A$ so that $G/A$ is a $\Sigma $-group, then $G$ is a $\Sigma $-group if and only if $A$ is a $\Sigma $-subgroup in $G$ provided that $A$ is equipped with a valuation induced by the restricted height function on $G$. In particular, if in addition $A$ is pure in $G$, $G$ is a $\Sigma $-group precisely when $A$ is a $\Sigma $-group. This extends the classical Dieudonné criterion (Portugal. Math., 1952) as well as it supplies our recent results in (Arch. Math. Brno, 2005), (Bull. Math. Soc. Sc. Math. Roumanie, 2006) and (Acta Math. Sci., 2007).
LA - eng
KW - height valuation; valuated subgroups; countable unions of subgroups; $\Sigma $-groups; valuated -groups; height valuations; valuated subgroups; nice subgroups; pure subgroups; Abelian -groups; -groups
UR - http://eudml.org/doc/35157
ER -

References

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  1. Danchev P. V., Commutative group algebras of abelian Σ -groups, , Math. J. Okayama Univ. (2) 40 (1998), 77-90. (1998) MR1755921
  2. Danchev P. V., Countable extensions of torsion abelian groups, , Arch. Math. (Brno) (3) 41 (2005), 265-272. Zbl1114.20030MR2188382
  3. Danchev P. V., Generalized Dieudonné criterion, , Acta Math. Univ. Comenianae (1) 74 (2005), 15-24. Zbl1111.20045MR2154393
  4. Danchev P. V., The generalized criterion of Dieudonné for valuated abelian groups, , Bull. Math. Soc. Sc. Math. Roumanie (2) 49 (2006), 149-155. MR2223310
  5. Danchev P. V., The generalized criterion of Dieudonné for primary valuated groups, , Acta Math. Sci. (3) 27 (2007). Zbl1197.20047MR2418776
  6. Danchev P. V., Generalized Dieudonné and Hill criteria, , Portugal. Math. (1) 64 (2007). Zbl1146.20034MR2418776
  7. Danchev P. V., Generalized Dieudonné and Honda criteria, , to appear. Zbl1166.20047MR2477120
  8. Dieudonné J. A., Sur les p -groupes abéliens infinis, , Portugal. Math. (1) 11 (1952), 1-5. (1952) MR0046356
  9. Fuchs L., Infinite Abelian Groups, , volumes I and II, Mir, Moskva 1974 and 1977 (in Russian). (1974) Zbl0338.20063MR0457533
  10. Kulikov L. Y., On the theory of abelian groups with arbitrary cardinality I and II, , Mat. Sb. 9 (1941), 165-182 and 16 (1945), 129-162 (in Russian). (1941) MR0018180

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