On deviation in groups
The author defines (by analogy with rings) the Krull dimension of a group. Let \(A\) be a partially ordered set. If \(a,b\in A\) then define the factor \(a/b\) as the subset \(\{x\in A\mid a\leq x\leq b\}\). The deviation of \(A\) (\(\text{dev }A\)) is defined as follows. If \(A\) is trivial, then \(\text{dev }A=-\infty\). If \(A\) is nontrivial but satisfies the minimal condition then \(\text{dev }A=0\). For a general ordinal \(\alpha\), we define \(\text{dev }A=\alpha\) provided: (1) \(\text{dev }A\not=\beta<\alpha\), (2) in any descending chain of elements of \(A\) all but not finitely many factors have deviation less than \(\alpha\) [\textit{J. C. McConnell} and \textit{J. C. Robson}, Noncommutative Noetherian rings, Wiley, New York (1987; Zbl 0980.16019), 6.1.1]. It should be noted that a partially ordered set need not have a deviation. Let \(G\) be a group. If \(A\) is the set of all normal subgroups of \(G\) then \(\text{dev }A\) is called the Krull dimension of the group \(G\). Let \(G\) be a group, \(H\) a normal subgroup of \(G\). We say that \(H\) satisfies the condition Max-\(G\) (respectively Min-\(G\)) if \(H\) satisfies the maximal (respectively minimal) condition for \(G\)-invariant subgroups. A group \(G\) is said to be generalized minimax, if it has a finite series of normal subgroups every factor of which is Abelian and satisfies Max-\(G\) or Min-\(G\). The generalized minimax groups were introduced in the paper [\textit{L. A. Kurdachenko}, \textit{J. Otal} and \textit{I. Ya. Subbotin}, On some criteria of nilpotency, Commun. Algebra 30, No. 8, 3755-3776 (2002; Zbl 1062.20041)]. The main result of this paper is Theorem 4.5. Let \(G\) be a metabelian group. The Krull dimension of \(G\) exists if and only if \(G\) is a generalized minimax group.
scientific article; zbMATH DE number 1993652
scientific article