The seventh and eighth Lie dimension subgroups. (English) Zbl 0809.20001
Let \(ZG\) be the integral group ring of a group \(G\) and let \(\Delta^{(n)} (G)\) denote the \(n\)-th Lie power of the augmentation ideal \(\Delta (G)\) of \(ZG\). The intersection \(D_{(n)} (G) = G \cap (1 + \Delta^{(n)}(G))\) is known as the \(n\)-th Lie dimension subgroup of \(G\). The authors prove that for \(n = 7\) and \(n = 8\) the \(n\)-th Lie dimension subgroup \(D_{(n)} (G)\) coincides with the \(n\)-th lower central subgroup \(\gamma_ n(G)\) for any group \(G\). R. Sandling [J. Algebra 21, 216-231 (1972; Zbl 0233.20001)] has proved that \(D_{(n)}(G) = \gamma_ n(G)\) for \(n \leq 6\). On the other hand, T. C. Hurley and S. K. Sehgal [ibid. 143, 46-56 (1991; Zbl 0761.20003)] have shown that there exists a 2-group \(G\) such that \(D_{(n)} (G) \neq \gamma_ n(G)\) for \(n \geq 9\). Thus, this paper completely resolves the Lie dimension subgroup conjecture. Together with the known other results, this resolves also the restricted Lie dimension subgroup conjecture for all values of \(n\).
Reviewer: S.V.Mihovski (Plovdiv)
MSC:
20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
20F14 | Derived series, central series, and generalizations for groups |
20F40 | Associated Lie structures for groups |
20E07 | Subgroup theorems; subgroup growth |
20F12 | Commutator calculus |
16S34 | Group rings |
Keywords:
integral group ring; Lie power; augmentation ideal; Lie dimension subgroup; lower central subgroup; restricted Lie dimension subgroup conjectureReferences:
[1] | Gupta, N., Free Group Rings, (Contemporary Mathematics, Vol. 66 (1987), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0641.20022 |
[2] | Gupta, N., The dimension subgroup conjecture, Bull. London Math. Soc., 22, 453-456 (1990) · Zbl 0721.20001 |
[3] | Gupta, N., On groups without dimension property, Internat. J. Algebra Comput., 1, 247-252 (1991) · Zbl 0781.20004 |
[4] | Gupta, N.; Srivastava, J. B., Some remarks on Lie dimension subgroups, J. Algebra, 143, 57-62 (1991) · Zbl 0761.20004 |
[5] | Hurley, T. C.; Sehgal, S. K., The Lie dimension subgroup conjecture, J. Algebra, 143, 46-56 (1991) · Zbl 0761.20003 |
[6] | Passi, I. B.S., Group Rings and Their Augmentation Ideals, (Lecture Notes in Mathematics, Vol. 715 (1970), Springer: Springer Berlin) · Zbl 1028.20006 |
[7] | Sandling, R., The dimension subgroup problem, J. Algebra, 21, 216-231 (1972) · Zbl 0233.20001 |
[8] | K.-I. Tahara and J. Xiao, On the seventh Lie dimension subgroups, Japan J. Math., to appear.; K.-I. Tahara and J. Xiao, On the seventh Lie dimension subgroups, Japan J. Math., to appear. · Zbl 0945.20503 |
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