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The generalized criterion of Dieudonné for valuated $p$-groups
Communications in Mathematics, Tome 14 (2006) no. 1, pp. 17-19
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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 $\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).
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).
Classification : 20K10, 20K27
Keywords: height valuation; valuated subgroups; countable unions of subgroups; $\Sigma $-groups 
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Danchev, Peter. The generalized criterion of Dieudonné for valuated $p$-groups. Communications in Mathematics, Tome 14 (2006) no. 1, pp. 17-19. http://geodesic.mathdoc.fr/item/COMIM_2006_14_1_a2/

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