Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
countable quotient groups; $\omega $-elongations; $p^{\omega +n}$-totally projective groups; $p^{\omega +n}$-summable groups
countable quotient groups; $\omega $-elongations; $p^{\omega +n}$-totally projective groups; $p^{\omega +n}$-summable groups
Summary:
It is proved that if $A$ is an abelian $p$-group with a pure subgroup $G$ so that $A/G$ is at most countable and $G$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable, then $A$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable as well. Moreover, if in addition $G$ is nice in $A$, then $G$ being either strongly $p^{\omega +n}$-totally projective or strongly $p^{\omega +n}$-summable implies that so is $A$. This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective $p$-groups as well as continues our recent investigations in (Arch. Math. (Brno), 2005 and 2006). Some other related results are also established.
References:
[1] Benabdallah K., Eisenstadt B., Irwin J., Poluianov E.: The structure of large subgroups of primary abelian groups. Acta Math. Acad. Sci. Hungar. 21 (3-4) (1970), 421–435. MR 0276328 | Zbl 0215.39804
[2] Cutler D.: Quasi-isomorphism for infinite abelian $p$-groups. Pacific J. Math. 16 (1) (1966), 25–45. MR 0191954 | Zbl 0136.28904
[3] Danchev P.: Characteristic properties of large subgroups in primary abelian groups. Proc. Indian Acad. Sci. Math. Sci. 104 (3) (2004), 225–233. MR 2083463 | Zbl 1062.20059
[4] Danchev P.: Countable extensions of torsion abelian groups. Arch. Math. (Brno) 41 (3) (2005), 265–272. MR 2188382 | Zbl 1114.20030
[5] 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
[6] Danchev P.: Generalized Wallace theorems. submitted. Zbl 1169.20029
[7] Danchev P.: Theorems of the type of Cutler for abelian $p$-groups. submitted. Zbl 1179.20046
[8] Danchev P.: Commutative group algebras of summable $p$-groups. Comm. Algebra 35 (2007). MR 2313667 | Zbl 1122.20003
[9] Danchev P.: Invariant properties of large subgroups in abelian $p$-groups. Oriental J. Math. Sci. 1 (1) (2007). MR 2656103 | Zbl 1196.20060
[10] Fuchs L.: Infinite Abelian Groups. I and II, Mir, Moskva, 1974 and 1977 (in Russian). MR 0457533 | Zbl 0338.20063
[11] Fuchs L., Irwin J.: On elongations of totally projective $p$-groups by $p^{\omega +n}$-projective $p$-groups. Czechoslovak Math. J. 32 (4) (1982), 511–515. MR 0682128
[12] Nunke R.: Homology and direct sums of countable abelian groups. Math. Z. 101 (3) (1967), 182–212. MR 0218452 | Zbl 0173.02401
[13] Nunke R.: Uniquely elongating modules. Symposia Math. 13 (1974), 315–330. MR 0364491 | Zbl 0338.20018
[14] Wallace K.: On mixed groups of torsion-free rank one with totally projective primary components. J. Algebra 17 (4) (1971), 482–488. MR 0272891 | Zbl 0215.39902
Search
Partner of
