Modules over nilpotent groups of finite rank
This paper arises from the study of various minimality conditions on infinite soluble groups. As usual, this leads to interesting questions about modules over group rings. In this case, the groups in question are nilpotent. Most of the work in the paper is module theoretic, and the results obtained are applied to obtain information about weak minimality conditions. The minimality condition of principal interest here is Min-\(\infty\)-n, a weak version of Min-n, the minimal condition for normal subgroups. For an account of locally soluble groups with Min-n, see \textit{J. S. Wilson} [Proc. Miniconf., Theory of Groups, Canberra 1975, Lect. Notes Math. 573, 130-142 (1977; Zbl 0361.20041)]. For the moment we note that soluble groups with Min-n are periodic (Baer), metabelian ones are countable (McDougall), while soluble groups of derived length three need not be countable (the reviewer). The weak minimality conditions Min-\(\infty\) and Min-\(\infty\)-n are defined as follows. A group G satisfies Min-\(\infty\) (resp. Min-\(\infty\)- n) if in any descending chain \(G_ 1\geq G_ 2\geq..\). of subgroups (resp. normal subgroups) of G, the indices \(| G_ i:G_{i+1}|\) are all finite beyond some point. These conditions have received some attention recently. Analogously to the above, the following results are obtained. Every metabelian group satisfying Min-\(\infty\)-n, is countable (Corollary to Theorem 4.2). Theorem 4.3: For torsion-free metanilpotent groups, the conditions Min-\(\infty\)-n and Min-\(\infty\) coincide, and hence torsionfree metabelian groups with Min-\(\infty\)-n are minimax. Theorem 4.4, which has a fairly straightforward self-contained proof, states that for periodic soluble groups, Min-\(\infty\)-n and Min-n are equivalent. With the exception of Theorem 4.4, these results are deduced from module theoretic theorems. These deal with JG-modules, where J is \({\mathbb{Z}}\) or the group ring \(F<t>\) of an infinite cyclic group over a finite field F, and G is a nilpotent group of finite free rank, meaning that the quotient of G by its torsion subgroup has finite rank. Let \(\sigma\) (J) be a set of representatives for the associate classes of primes of J. Theorem 3.1: Let A be a finitely generated JG-module, and if \(J=F<t>\), assume that the characteristic of F is not in the spectrum of G (see below). If A has infinite J-rank, then A/Au is infinite for all but finitely many \(u\in \sigma (J)\). The spectrum is defined as follows. Let A be any J-module and \(A_ 0\) be the submodule generated by a maximal J- linearly independent set. We say that the prime u belongs to the spectrum \(Sp_ JA\), if the u-primary component of \(A/A_ 0\) is infinite. This is independent of the choice of \(A_ 0\), provided that \(A_ 0\) can be chosen of finite rank. The spectrum Sp G of a nilpotent group G of finite free rank is then defined to be the union of the spectra of the factors in some central series of G. Theorem 3.1 breaks down if the condition on the characteristic of F is omitted, but the following is stated as a replacement. Theorem 3.3: Let A be a noetherian JG-module, where \(J=F<t>\), and suppose A has infinite J-rank. Then A/Au is infinite for all but finitely many \(u\in \sigma (J)\). The reviewer has been unable to understand the proof of this theorem. In the paragraph following (3.1), it is stated that A can be viewed as an F-algebra. This seems to amount to viewing JG/R as an F-algebra where R is a right, but not two-sided, ideal of JG. Further, the argument that certain elements have p-powers in E seems to assume \((g+h)^ p=g^ p+h^ p\) where g and h need not commute. For the group theoretical applications, Theorem 3.3 appears to be used only in proving Theorem 4.1: If A is a minimally infinite \({\mathbb{Z}}G\)- module, in the sense that every proper quotient is finite and the non- zero submodules have trivial intersection, and G is nilpotent of finite free rank, then \(G/C_ G(A)\) is finitely generated and its centre has finite index. This can be proved by combining arguments of the present paper with an argument of \textit{C. J. B. Brookes} [Bull. Lond. Math. Soc. 18, 7-10 (1986; Zbl 0556.20028), p. 9]. As far as the reviewer is aware, the other results in this paper are unaffected, including the interesting Theorem 4.2: Let A be a \({\mathbb{Z}}G\)-module satisfying Min-\(\infty\) on submodules, where G is nilpotent of finite free rank. Then G is minimax (as \({\mathbb{Z}}G\)-module).
minimality conditions on infinite soluble groups
scientific article; zbMATH DE number 3975210