# Article

Full entry | PDF   (0.4 MB)
Keywords:
abelian groups; countable factor-groups; $p^{\omega +n}$-projective groups
Summary:
We prove that if $G$ is an Abelian $p$-group of length not exceeding $\omega$ and $H$ is its $p^{\omega +n}$-projective subgroup for $n\in {\mathbb{N}} \cup \lbrace 0\rbrace$ such that $G/H$ is countable, then $G$ is also $p^{\omega +n}$-projective. This enlarges results of ours in (Arch. Math. (Brno), 2005, 2006 and 2007) as well as a classical result due to Wallace (J. Algebra, 1971).
References:
[1] Danchev, P.: Countable extensions of torsion abelian groups. Arch. Math. (Brno) 41 (3) (2005), 265–272. MR 2188382 | Zbl 1114.20030
[2] Danchev, P.: Generalized Dieudonné criterion. Acta Math. Univ. Comenian. 74 (1) (2005), 15–26. MR 2154393 | Zbl 1111.20045
[3] Danchev, P.: A note on the countable extensions of separable $p^{\omega +n}$-projective abelian $p$-groups. Arch. Math. (Brno) 42 (3) (2006), 251–254. MR 2260384 | Zbl 1152.20045
[4] Danchev, P.: On countable extensions of primary abelian groups. Arch. Math. (Brno) 43 (1) (2007), 61–66. MR 2310125 | Zbl 1156.20044
[5] Danchev, P., Keef, P.: Generalized Wallace theorems. Math. Scand. (to appear). MR 2498370
[6] Dieudonné, J.: Sur les $p$-groupes abeliens infinis. Portugal. Math. 11 (1) (1952), 1–5. MR 0046356
[7] Fuchs, L.: Infinite Abelian Groups, I, II. Mir, Moskva, 1974 and 1977, (in Russian). MR 0457533
[8] Fuchs, L.: Subfree valued vector spaces. Lecture Notes in Math. 616 (1977), 158–167. DOI 10.1007/BFb0068194 | MR 0480700 | Zbl 0389.15001
[9] Hill, P., Megibben, C.: Extensions of torsion-complete groups. Proc. Amer. Math. Soc. 44 (2) (1974), 259–262. DOI 10.1090/S0002-9939-1974-0340452-3 | MR 0340452 | Zbl 0292.20050
[10] Nunke, R.: Topics in Abelian Groups. ch. Purity and subfunctors of the identity, pp. 121–171, Scott, Foresman and Co., Chicago, 1963. MR 0169913
[11] Nunke, R.: Homology and direct sums of countable abelian groups. Math. Z. 101 (3) (1967), 182–212. DOI 10.1007/BF01135839 | MR 0218452 | Zbl 0173.02401
[12] Wallace, K.: On mixed groups of torsion-free rank one with totally projective primary components. J. Algebra 17 (4) (1971), 482–488. DOI 10.1016/0021-8693(71)90005-6 | MR 0272891 | Zbl 0215.39902

Partner of