Modular group algebras of ℵ₁-separable 𝑝-groups

Göbel, Rüdiger; May, Warren

doi:10.1090/S0002-9939-03-06818-7

Modular group algebras of $\aleph _{1}$-separable $p$-groups
HTML articles powered by AMS MathViewer

by Rüdiger Göbel and Warren May PDF

Proc. Amer. Math. Soc. 131 (2003), 2987-2992 Request permission

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

S. D. Berman, Group algebras of countable abelian $p$-groups, Publ. Math. Debrecen 14 (1967), 365–405 (Russian). MR 225897
S. D. Berman and T. Ž. Mollov, The group rings of abelian $p$-groups of arbitrary power, Mat. Zametki 6 (1969), 381–392 (Russian). MR 254155
Peter Crawley, The cancellation of torsion abelian groups in direct sums, J. Algebra 2 (1965), 432–442. MR 188293, DOI 10.1016/0021-8693(65)90004-9
Paul C. Eklof, The structure of $\omega _{1}$-separable groups, Trans. Amer. Math. Soc. 279 (1983), no. 2, 497–523. MR 709565, DOI 10.1090/S0002-9947-1983-0709565-8
Paul C. Eklof and Alan H. Mekler, On endomorphism rings of $\omega _{1}$-separable primary groups, Abelian group theory (Honolulu, Hawaii, 1983) Lecture Notes in Math., vol. 1006, Springer, Berlin, 1983, pp. 320–339. MR 722627, DOI 10.1007/BFb0103711
Paul C. Eklof and Alan H. Mekler, Almost free modules, North-Holland Mathematical Library, vol. 46, North-Holland Publishing Co., Amsterdam, 1990. Set-theoretic methods. MR 1055083
Walter Leighton and W. T. Scott, A general continued fraction expansion, Bull. Amer. Math. Soc. 45 (1939), 596–605. MR 41, DOI 10.1090/S0002-9904-1939-07046-8
R. Göbel, W. May, Cancellation of direct sums of countable abelian $p$-groups, to appear.
Paul Hill and William Ullery, A note on a theorem of May concerning commutative group algebras, Proc. Amer. Math. Soc. 110 (1990), no. 1, 59–63. MR 1039530, DOI 10.1090/S0002-9939-1990-1039530-4
Martin Huber, Methods of set theory and the abundance of separable abelian $p$-groups, Abelian group theory (Honolulu, Hawaii, 1983) Lecture Notes in Math., vol. 1006, Springer, Berlin, 1983, pp. 304–319. MR 722626, DOI 10.1007/BFb0103710
Gregory Karpilovsky, Commutative group algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 78, Marcel Dekker, Inc., New York, 1983. MR 704185
Warren May, Commutative group algebras, Trans. Amer. Math. Soc. 136 (1969), 139–149. MR 233903, DOI 10.1090/S0002-9947-1969-0233903-9
Warren May, The direct factor problem for modular abelian group algebras, Representation theory, group rings, and coding theory, Contemp. Math., vol. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 303–308. MR 1003359, DOI 10.1090/conm/093/1003359