Geodesic
Isotype knice subgroups of global Warfield groups
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 109-132 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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If $H$ is an isotype knice subgroup of a global Warfield group $G$, we introduce the notion of a $k$-subgroup to obtain various necessary and sufficient conditions on the quotient group $G/H$ in order for $H$ itself to be a global Warfield group. Our main theorem is that $H$ is a global Warfield group if and only if $G/H$ possesses an $H(\aleph _0)$-family of almost strongly separable $k$-subgroups. By an $H(\aleph _0)$-family we mean an Axiom 3 family in the strong sense of P. Hill. As a corollary to the main theorem, we are able to characterize those global $k$-groups of sequentially pure projective dimension $\le 1$.
If $H$ is an isotype knice subgroup of a global Warfield group $G$, we introduce the notion of a $k$-subgroup to obtain various necessary and sufficient conditions on the quotient group $G/H$ in order for $H$ itself to be a global Warfield group. Our main theorem is that $H$ is a global Warfield group if and only if $G/H$ possesses an $H(\aleph _0)$-family of almost strongly separable $k$-subgroups. By an $H(\aleph _0)$-family we mean an Axiom 3 family in the strong sense of P. Hill. As a corollary to the main theorem, we are able to characterize those global $k$-groups of sequentially pure projective dimension $\le 1$.
Classification : 20K21, 20K27
Keywords: global Warfield group; isotype subgroup; knice subgroup; $k$-subgroup; separable subgroup; compatible subgroups; Axiom 3; closed set method; global $k$-group; sequentially pure projective dimension 
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Megibben, Charles; Ullery, William. Isotype knice subgroups of global Warfield groups. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 109-132. http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a8/

[1] U. Albrecht and P. Hill: Butler groups of infinite rank and axiom 3. Czechoslovak Math. J. 37 (1987), 293–309. | MR

[2] L. Fuchs: Infinite Abelian Groups. Vol. II, Academic Press, New York, 1973. | MR | Zbl

[3] L. Fuchs and P. Hill: The balanced projective dimension of abelian $p$-groups. Trans. Amer. Math. Soc. 293 (1986), 99–112. | MR

[4] L. Fuchs and M. Magidor: Butler groups of arbitrary cardinality. Israel J. Math. 84 (1993), 239–263. | DOI | MR

[5] P. Hill: The third axiom of countability for abelian groups. Proc. Amer. Math. Soc. 82 (1981), 347–350. | DOI | MR | Zbl

[6] P. Hill: Isotype subgroups of totally projective groups. Lecture Notes in Math vol. 874, Springer-Verlag, New York, 1981, pp. 305–321. | MR | Zbl

[7] P. Hill and C. Megibben: Knice subgroups of mixed groups. Abelian Group Theory, Gordon-Breach, New York, 1987, pp. 89–109. | MR

[8] P. Hill and C. Megibben: Mixed groups. Trans. Amer. Math. Soc. 334 (1992), 121–142. | DOI | MR

[9] P. Hill and C. Megibben: The nonseparability of simply presented mixed groups. Comment. Math. Univ. Carolinae 39 (1998), 1–5. | MR

[10] P. Hill and W. Ullery: Isotype subgroups of local Warfield groups. Comm. Algebra 29 (2001), 1899–1907. | MR

[11] C. Megibben and W. Ullery: Isotype subgroups of mixed groups. Comment. Math. Univ. Carolinae 42 (2001), 421–442. | MR

[12] C. Megibben and W. Ullery: Isotype Warfield subgroups of global Warfield groups. Rocky Mountain J. Math. 32 (2002), 1523–1542. | DOI | MR

[13] R. Warfield: Simply presented groups. Proc. Sem. Abelian Group Theory, Univ. of Arizona lecture notes, 1972.