A new class of abelian $p$-groups with all high subgroups isomorphic is defined. Commutative modular and semisimple group algebras over such groups are examined. The results obtained continue our recent statements published in Comment. Math. Univ. Carolinae (2002).
@article{119413, author = {Peter Vassilev Danchev}, title = {Commutative group algebras of highly torsion-complete abelian $p$-groups}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {44}, year = {2003}, pages = {587-592}, zbl = {1101.20001}, mrnumber = {2062875}, language = {en}, url = {http://dml.mathdoc.fr/item/119413} }
Danchev, Peter Vassilev. Commutative group algebras of highly torsion-complete abelian $p$-groups. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 587-592. http://gdmltest.u-ga.fr/item/119413/
Primary abelian groups having all high subgroups isomorphic, Proc. Amer. Math. Soc. (3) 83 (1981), 467-470. (1981) | MR 0627671
High subgroups of primary abelian groups of length $ømega+n$, J. Algebra 83 (1983), 502-509. (1983) | MR 0714261
Normed unit groups and direct factor problem for commutative modular group algebras, Math. Balkanica (2-3) 10 (1996), 161-173. (1996) | MR 1606535 | Zbl 0980.16500
Isomorphism of commutative semisimple group algebras, Math. Balkanica (1-2) 11 (1997), 51-55. (1997) | MR 1606596 | Zbl 0939.16015
Isomorphism of modular group algebras of direct sums of torsion-complete abelian $p$-groups, Rend. Sem. Mat. Univ. Padova 101 (1999), 51-58. (1999) | MR 1705279 | Zbl 0959.20003
Commutative group algebras of cardinality $\aleph_1$, Southeast Asian Bull. Math. (4) 25 (2001), 589-598. (2001) | MR 1934659
Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma$-groups, Comment. Math. Univ. Carolinae 43 (2002), 419-428. (2002) | MR 1920518 | Zbl 1068.16042
Invariants for group algebras of abelian groups, Rend. Circolo Mat. Palermo (2) 51 (2002), 391-402. (2002) | MR 1947462 | Zbl 1097.20504
Isomorphism of commutative group algebras of closed $p$-groups and $p$-local algebraically compact groups, Proc. Amer. Math. Soc. 130 (2002), 1937-1941. (2002) | MR 1896025 | Zbl 0997.20007
Sylow $p$-subgroups of abelian group rings, Serdica Math. J. 29 (2003), 33-44. (2003) | MR 1981103 | Zbl 1035.16025
Isomorphic commutative group algebras of primary semi-complete abelian groups, to appear.
Torsion completeness of Sylow $p$-groups in semi-simple group rings, Acta Math. Sinica (2003). (2003)
Commutative group algebras of $p^{ømega+n}$-projective abelian groups, submitted. | Zbl 1111.20007
$S(RG)/G_p$ is a $\Sigma$-group, in press.
Infinite Abelian Groups, I and II, Mir, Moscow, 1974 and 1977 (in Russian). | MR 0457533 | Zbl 0338.20063
The equivalence of high subgroups, Proc. Amer. Math. Soc. 88 (1983), 207-211. (1983) | MR 0695242 | Zbl 0522.20038
High subgroups of abelian torsion groups, Pacific J. Math. 11 (1961), 1375-1384. (1961) | MR 0136654 | Zbl 0106.02303
A functorial generalization of Ulm's theorem, Comment. Math. Univ. St. Pauli 27 (1978), 155-177. (1978) | MR 0546205
Countable direct sums of torsion complete groups, Proc. Amer. Math. Soc. 17 (1966), 763-766. (1966) | MR 0197556 | Zbl 0143.04501
The torsion product of totally projective $p$-groups, Comment. Math. Univ. St. Pauli 29 (1980), 1-5. (1980) | MR 0586242 | Zbl 0441.20035
Sylow $p$-subgroups of the group of normed units of semisimple group algebras of uncountable abelian $p$-groups, Pliska Stud. Math. Bulgar. 8 (1986), 34-46 (in Russian). (1986) | MR 0866643