Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Modular group algebras of $\aleph_{1}$-separable $p$-groups


Authors: Rüdiger Göbel and Warren May
Journal: Proc. Amer. Math. Soc. 131 (2003), 2987-2992
MSC (2000): Primary 20K10, 20C07; Secondary 20K25, 16S34
DOI: https://doi.org/10.1090/S0002-9939-03-06818-7
Published electronically: January 2, 2003
MathSciNet review: 1993203
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Under the assumptions of MA and $\neg$ CH, it is proved that if $F$ is a field of prime characteristic $p$ and $G$ is an $\aleph_{1}$-separable abelian $p$-group of cardinality $\aleph_{1}$, then an isomorphism of the group algebras $FG$ and $FH$ implies an isomorphism of $G$ and $H$.


References [Enhancements On Off] (What's this?)

  • [B] S. D. Berman, Group algebras of countable abelian $p$-groups, Publ. Math. Debrecen 14 (1967), 365-405. MR 37:1488
  • [BM] S. D. Berman, T. Z. Mollov, On group rings of abelian $p$-groups of any cardinality, Matem. Zametki 6 (1969), 381-392. MR 40:7365
  • [C] P. Crawley, The cancellation of torsion abelian groups in direct sums, J. Algebra 2 (1965), 432-442. MR 32:5732
  • [E] P. Eklof, The structure of $\omega_{1}$-separable groups, Trans. Amer. Math. Soc. 279 (1983), 497-523. MR 84k:03124
  • [EM1] P. Eklof, A. Mekler, On endomorphism rings of $\omega_{1}$-separable primary groups, pp. 320-339 in ``Abelian Group Theory,'' ed. by R. Göbel, L. Lady, A. Mader, Lecture Notes in Math., Vol. 1006, Springer, Berlin, New York 1983. MR 85i:20052
  • [EM2] P. Eklof, A. Mekler, Almost Free Modules, Set Theoretic Methods, North-Holland, Amsterdam 1990. MR 92e:20001
  • [Fu] L. Fuchs, Infinite Abelian Groups, Vol. I, Vol. II, Academic Press, New York 1970, 1973. MR 41:333; MR 50:2362
  • [GM] R. Göbel, W. May, Cancellation of direct sums of countable abelian $p$-groups, to appear.
  • [HU] P. Hill, W. Ullery, A note on a theorem of May concerning commutative group algebras, Proc. Amer. Math. Soc. 110 (1990), 59-63. MR 91b:20008
  • [H] M. Huber, Methods of set theory and the abundance of separable abelian $p$-groups, pp. 304-319 in ``Abelian Group Theory,'' ed. by R. Göbel, L. Lady, A. Mader, Lecture Notes in Math., Vol. 1006, Springer, Berlin, New York 1983. MR 85h:20059
  • [K] G. Karpilovsky, Commutative Group Algebras, Pure and Appl. Math. Monographs, Vol. 78, Marcel Dekker, New York 1983. MR 85a:16014
  • [M1] W. May, Commutative group algebras, Trans. Amer. Math. Soc. 136 (1969), 139-149. MR 38:2224
  • [M] W. May, The direct factor problem for modular abelian group algebras, Contemp. Math. 93 (1989), 303-308. MR 90c:20071

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20K10, 20C07, 20K25, 16S34

Retrieve articles in all journals with MSC (2000): 20K10, 20C07, 20K25, 16S34


Additional Information

Rüdiger Göbel
Affiliation: Fachbereich 6, Mathematik und Informatik, Universität Essen, Universitätsstraße 3, D-45117 Essen, Germany
Email: R.Goebel@uni-essen.de

Warren May
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
Email: may@math.arizona.edu

DOI: https://doi.org/10.1090/S0002-9939-03-06818-7
Received by editor(s): January 25, 2002
Received by editor(s) in revised form: April 18, 2002
Published electronically: January 2, 2003
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society