Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
countable factor-groups; $\Sigma $-groups; $\sigma $-summable groups; summable groups; $p^{\omega + n}$-projective groups
countable factor-groups; $\Sigma $-groups; $\sigma $-summable groups; summable groups; $p^{\omega + n}$-projective groups
Summary:
Suppose $A$ is an abelian torsion group with a subgroup $G$ such that $A/G$ is countable that is, in other words, $A$ is a torsion countable abelian extension of $G$. A problem of some group-theoretic interest is that of whether $G \in \mathbb K$, a class of abelian groups, does imply that $A\in \mathbb K$. The aim of the present paper is to settle the question for certain kinds of groups, thus extending a classical result due to Wallace (J. Algebra, 1981) proved when $\mathbb K$ coincides with the class of all totally projective $p$-groups.
References:
[1] Danchev P. V.: Commutative group algebras of $\sigma $-summable abelian groups. Proc. Amer. Math. Soc. (9) 125 (1997), 2559–2564. MR 1415581 | Zbl 0886.16024
[2] Danchev P. V.: Commutative group algebras of abelian $\Sigma $-groups. Math. J. Okayama Univ. 40 (1998), 77–90. MR 1755921
[3] Danchev P. V.: Commutative group algebras of highly torsion-complete abelian $p$-groups. Comment. Math. Univ. Carolin. (4) 44 (2003), 587–592. MR 2062875 | Zbl 1101.20001
[4] Danchev P. V.: Commutative group algebras of summable abelian $p$-groups. Comm. Algebra, in press.
[5] Danchev P. V.: Generalized Dieudonné criterion. Acta Math. Univ. Comenian. (1) 74 (2005), 15–24. MR 2154393 | Zbl 1111.20045
[6] Fuchs L.: Infinite Abelian Groups. Volumes I and II, Mir, Moskva, 1974 and 1977. (In Russian.) MR 0457533 | Zbl 0338.20063
[7] Hill P. D.: Criteria for freeness in groups and valuated vector spaces. Lect. Notes in Math. 616 (1977), 140–157. MR 0486206 | Zbl 0372.20041
[8] Hill P. D., Megibben C. K.: On direct sums of countable groups and generalizations. Études sur les Groupes Abéliens, Paris (1968), 183–206. MR 0242943 | Zbl 0203.32705
[9] Hill P. D., Megibben C. K.: Extending automorphisms and lifting decompositions in abelian groups. Math. Annalen 175 (1968), 159–168. MR 0223449 | Zbl 0183.03202
[10] Megibben C. K.: The generalized Kulikov criterion. Canad. J. Math. 21 (1969), 1192–1205. MR 0249509 | Zbl 0208.03502
[11] Megibben C. K.: Countable extensions of simply presented groups. Internet information.
[12] Wallace K. D.: On mixed groups of torsion-free rank one with totally projective primary components. J. Algebra 17 (1971), 482–488. MR 0272891 | Zbl 0215.39902
[13] Nunke R. J.: Purity and subfunctors of the identity. Topics in Abelian Groups, Scott Foresman and Co., (1963), 121–171. MR 0169913
Search
Partner of
