Polynomial identities in graded group rings, restricted Lie algebras and 𝑝-adic analytic groups

Shalev, Aner

doi:10.1090/S0002-9947-1993-1093218-X

Polynomial identities in graded group rings, restricted Lie algebras and $p$-adic analytic groups
HTML articles powered by AMS MathViewer

by Aner Shalev PDF

Trans. Amer. Math. Soc. 337 (1993), 451-462 Request permission

Abstract:

Let $G$ be any finitely generated group, and let $K$ be a field of characteristic $p > 0$. It is shown that the graded group ring $\operatorname {gr}(KG)$ satisfies a nontrivial polynomial identity if and only if the pro-$p$ completion of $G$ is $p$-adic analytic, i.e. can be given the structure of a Lie group over the $p$-adic field ${\mathbb {Q}_p}$. The proof applies theorems of Lazard, Quillen and Passman, as well as results on Engel Lie algebras and on dimension subgroups in positive characteristic.

References

J. A. Bahturin, Lectures on Lie algebras, Studien zur Algebra und ihre Anwendungen [Studies in Algebra and its Applications], vol. 4, Akademie-Verlag, Berlin, 1978. Lectures given at Humboldt University, Berlin and Lomonosov University, Moscow. MR 526403
Marcus P. F. du Sautoy, Finitely generated groups, $p$-adic analytic groups and Poincaré series, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 121–126. MR 1021793, DOI 10.1090/S0273-0979-1990-15916-6

Analytic pro

groups

K. W. Gruenberg, Two theorems on Engel groups, Proc. Cambridge Philos. Soc. 49 (1953), 377–380. MR 55347, DOI 10.1017/s0305004100028528
I. M. Isaacs and D. S. Passman, Groups with representations of bounded degree, Canadian J. Math. 16 (1964), 299–309. MR 167539, DOI 10.4153/CJM-1964-029-5
S. A. Jennings, The structure of the group ring of a $p$-group over a modular field, Trans. Amer. Math. Soc. 50 (1941), 175–185. MR 4626, DOI 10.1090/S0002-9947-1941-0004626-6
S. A. Jennings, The group ring of a class of infinite nilpotent groups, Canadian J. Math. 7 (1955), 169–187. MR 68540, DOI 10.4153/CJM-1955-022-5
V. N. Latyšev, Two remarks on $PI$-algebras, Sibirsk. Mat. Ž. 4 (1963), 1120–1121 (Russian). MR 0156873
Michel Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101–190 (French). MR 0088496

Groupes analytiques

adiques

26

Alexander Lubotzky, A group theoretic characterization of linear groups, J. Algebra 113 (1988), no. 1, 207–214. MR 928062, DOI 10.1016/0021-8693(88)90190-1

Pro

groups of finite coclass

Alexander Lubotzky and Avinoam Mann, Powerful $p$-groups. I. Finite groups, J. Algebra 105 (1987), no. 2, 484–505. MR 873681, DOI 10.1016/0021-8693(87)90211-0
Alexander Lubotzky and Avinoam Mann, Powerful $p$-groups. II. $p$-adic analytic groups, J. Algebra 105 (1987), no. 2, 506–515. MR 873682, DOI 10.1016/0021-8693(87)90212-2
Alexander Lubotzky and Avinoam Mann, Residually finite groups of finite rank, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 3, 385–388. MR 1010362, DOI 10.1017/S0305004100068110
Alexander Lubotzky and Avinoam Mann, On groups of polynomial subgroup growth, Invent. Math. 104 (1991), no. 3, 521–533. MR 1106747, DOI 10.1007/BF01245088

Some applications of powerful

groups

Avinoam Mann and Dan Segal, Uniform finiteness conditions in residually finite groups, Proc. London Math. Soc. (3) 61 (1990), no. 3, 529–545. MR 1069514, DOI 10.1112/plms/s3-61.3.529
D. S. Passman, Group rings satisfying a polynomial identity, J. Algebra 20 (1972), 103–117. MR 285631, DOI 10.1016/0021-8693(72)90091-9
D. S. Passman, Enveloping algebras satisfying a polynomial identity, J. Algebra 134 (1990), no. 2, 469–490. MR 1074338, DOI 10.1016/0021-8693(90)90062-S
V. M. Petrogradski, Identities in the enveloping algebras for modular Lie superalgebras, J. Algebra 145 (1992), no. 1, 1–21. MR 1144655, DOI 10.1016/0021-8693(92)90173-J
Daniel G. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411–418. MR 231919, DOI 10.1016/0021-8693(68)90069-0
Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
Aner Shalev, Dimension subgroups, nilpotency indices, and the number of generators of ideals in $p$-group algebras, J. Algebra 129 (1990), no. 2, 412–438. MR 1040946, DOI 10.1016/0021-8693(90)90228-G
Aner Shalev, Characterization of $p$-adic analytic groups in terms of wreath products, J. Algebra 145 (1992), no. 1, 204–208. MR 1144667, DOI 10.1016/0021-8693(92)90185-O
Carlo M. Scoppola and Aner Shalev, Applications of dimension subgroups to the power structure of $p$-groups, Israel J. Math. 73 (1991), no. 1, 45–56. MR 1119926, DOI 10.1007/BF02773423