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Keywords:
height valuation; valuated subgroups; countable unions of subgroups; $\Sigma $-groups
height valuation; valuated subgroups; countable unions of subgroups; $\Sigma $-groups
Summary:
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).
References:
[1] Danchev P. V.: Commutative group algebras of abelian $\Sigma $-groups. Math. J. Okayama Univ. (2) 40 (1998), 77-90. MR 1755921
[2] Danchev P. V.: Countable extensions of torsion abelian groups. Arch. Math. (Brno) (3) 41 (2005), 265-272. MR 2188382 | Zbl 1114.20030
[3] Danchev P. V.: Generalized Dieudonné criterion. Acta Math. Univ. Comenianae (1) 74 (2005), 15-24. MR 2154393 | Zbl 1111.20045
[4] Danchev P. V.: The generalized criterion of Dieudonné for valuated abelian groups. Bull. Math. Soc. Sc. Math. Roumanie (2) 49 (2006), 149-155. MR 2223310
[5] Danchev P. V.: The generalized criterion of Dieudonné for primary valuated groups. Acta Math. Sci. (3) 27 (2007). MR 2418776 | Zbl 1197.20047
[6] Danchev P. V.: Generalized Dieudonné and Hill criteria. Portugal. Math. (1) 64 (2007). MR 2418776 | Zbl 1146.20034
[7] Danchev P. V.: Generalized Dieudonné and Honda criteria. to appear. MR 2477120 | Zbl 1166.20047
[8] Dieudonné J. A.: Sur les $p$-groupes abéliens infinis. Portugal. Math. (1) 11 (1952), 1-5. MR 0046356
[9] Fuchs L.: Infinite Abelian Groups. volumes I and II, Mir, Moskva 1974 and 1977 (in Russian). MR 0457533 | Zbl 0338.20063
[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). MR 0018180
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