Group rings with solvable 𝑛-Engel unit groups

Fisher, J. L.; Parmenter, M. M.; Sehgal, S. K.

doi:10.1090/S0002-9939-1976-0417231-3

Group rings with solvable $n$-Engel unit groups
HTML articles powered by AMS MathViewer

by J. L. Fisher, M. M. Parmenter and S. K. Sehgal PDF

Proc. Amer. Math. Soc. 59 (1976), 195-200 Request permission

Abstract:

Let $KG$ be the group ring of a group $G$ over a field of characteristic $p \geqslant 0,\;p \ne 2,\;3$. Suppose $G$ contains no element of order $p$ (if $p > 0$). Group algebras $KG$ with unit group $U(KG)$ solvable and $n$-Engel are characterized.

References

J. M. Bateman, On the solvability of unit groups of group algebras, Trans. Amer. Math. Soc. 157 (1971), 73–86. MR 276371, DOI 10.1090/S0002-9947-1971-0276371-2
J. M. Bateman and D. B. Coleman, Group algebras with nilpotent unit groups, Proc. Amer. Math. Soc. 19 (1968), 448–449. MR 222186, DOI 10.1090/S0002-9939-1968-0222186-6
A. A. Bovdi and S. V. Mihovski, Idempotents of crossed products, Dokl. Akad. Nauk SSSR 195 (1970), 263–265. MR 0272912
Loo-Keng Hua, On the multiplicative group of a field, Acad. Sinica Science Record 3 (1950), 1–6 (English, with Chinese summary). MR 0039707
I. I. Hripta, The nilpotence of the multiplicative group of a group ring, Mat. Zametki 11 (1972), 191–200 (Russian). MR 311750
Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
Charles Lanski, Some remarks on rings with solvable units, Ring theory (Proc. Conf., Park City, Utah, 1971) Academic Press, New York, 1972, pp. 235–240. MR 0342565
Charles Lanski, Subgroups and conjugates in semi-prime rings, Math. Ann. 192 (1971), 313–327. MR 286840, DOI 10.1007/BF02075359
A. Meir, An interpolatory rational approximation, Canad. Math. Bull. 21 (1978), no. 2, 197–200. MR 493045, DOI 10.4153/CMB-1978-033-6
Kaoru Motose and Hisao Tominaga, Group rings with solvable unit groups, Math. J. Okayama Univ. 15 (1971/72), 37–40. MR 306297
I. B. S. Passi, D. S. Passman, and S. K. Sehgal, Lie solvable group rings, Canadian J. Math. 25 (1973), 748–757. MR 325746, DOI 10.4153/CJM-1973-076-4
Donald S. Passman, Infinite group rings, Pure and Applied Mathematics, vol. 6, Marcel Dekker, Inc., New York, 1971. MR 0314951
Derek J. S. Robinson, Finiteness conditions and generalized soluble groups. Part 1, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 62, Springer-Verlag, New York-Berlin, 1972. MR 0332989
S. K. Sehgal, Lie properties in modular group algebras, Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972) Lecture Notes in Math., Vol. 353, Springer, Berlin, 1973, pp. 152–160. MR 0340396
W. Sierpiński, Elementary theory of numbers, 2nd ed., North-Holland Mathematical Library, vol. 31, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw, 1988. Edited and with a preface by Andrzej Schinzel. MR 930670