The associated graded ring of an integral group ring. (English) Zbl 0358.16006

Passi, Inder Bir S.; Vermani, Lekh Raj

Math. Proc. Camb. Philos. Soc. 82, 25-33 (1977).

Let \(G\) be an abelian group, \(S(G)\) the symmetric algebra of \(G\) and \(gr ZG\) the graded ring associated to the filtration of the ring \(ZG\) by powers of the augmentation ideal \(A_G\). There is a natural homogeneous surjection \(\theta:S(G) \to gr ZG\), with \(n\)-th component given by \[ \theta_n(x_1 \overset \wedge\otimes \dots \overset \wedge\otimes x_n) = \prod_{i=1}^n (x_i-1) + A^{n+1}_G. \]
Several results on the kernel of \(\theta\) are obtained; the main result is a determination (in the case of finite \(p\)-groups) of when \(\theta_n\), is an isomorphism. The discussion is highly technical but interesting.

Reviewer: William H. Gustafson (Lubbock)

MSC:

16S34	Group rings
20C05	Group rings of finite groups and their modules (group-theoretic aspects)
15A69	Multilinear algebra, tensor calculus

References:

[1]	Vermani, Proc. Cambridge Philos. Soc. 68 pp 285– (1970)
[2]	DOI: 10.1080/00927877508822085 · Zbl 0326.20007 · doi:10.1080/00927877508822085
[3]	Serre, Multiplicit?s. Lecture Notes in Mathematics 11 (1965)
[4]	DOI: 10.1007/BF02566808 · Zbl 0259.20034 · doi:10.1007/BF02566808
[5]	Passi, Proc. Cambridge Philos. Soc. 66 pp 505– (1969)
[6]	DOI: 10.1016/0021-8693(68)90017-3 · Zbl 0159.31502 · doi:10.1016/0021-8693(68)90017-3
[7]	DOI: 10.1016/0022-4049(74)90036-X · Zbl 0301.16011 · doi:10.1016/0022-4049(74)90036-X
[8]	Sandling, Proc. Cambridge Philos. Soc. 71 pp 25– (1972)