Let \(\chi\) be a subgroup theoretical property. A group \(G\) is said to satisfy the condition max-\(\infty\chi\) if there are no infinite strictly ascending chains of infinite \(\chi\)-subgroups of \(G\); similarly, we shall say that \(G\) satisfies min-\(\infty\chi\) if there are no infinite strictly descending chains of \(\chi\)-subgroups of infinite index of \(G\). The structure of (soluble) groups for which this type of chain conditions are imposed on the set of all subgroups (max-\(\infty\) and min-\(\infty\)) or on set of all subnormal subgroups (max-\(\infty sn\) and min-\(\infty sn\)) was investigated by D. H. Paek (2001, 2002). Moreover, he also studied the locally nilpotent groups with these conditions for normal subgroups (max-\(\infty n\) and min-\(\infty n\)) (2004). Recently, F. de Giovanni, D. H. Paek, D. J. S. Robinson and A. Russo [Commun. Algebra 33, No. 1, 183-199 (2005; Zbl 1070.20034)] have examined the structure of groups which satisfy max-\(\infty n\) and min-\(\infty n\) given characterizations of both properties in the general case. Moreover, detailed information is obtained about soluble groups with these chain conditions.
In the article under review the author gives a description of metanilpotent groups satisfying the condition max-\(\infty n\) (respectively, the condition min-\(\infty n\)). He proves the following theorems:
1. Let \(G\) be a metanilpotent group, and put \(F=\text{Fit}(G)\) and \(Z=Z(G)\). Then \(G\) satisfies max-\(\infty n\) but not max iff the following hold: (1) \(F\) is infinite nilpotent and \(G/F\) has max-\(n\); (2) if \(L\) is an infinite normal subgroup of \(G\), then \(Z/Z\cap L\) is finite.
2. Let \(G\) be a metanilpotent group, and put \(F=\text{Fit}(G)\). Then \(G\) satisfies min-\(\infty n\) but not min iff \(F\) is infinite normal nilpotent and either \(F\) is a \(G\)-rationally irreducible free Abelian subgroup of finite rank such that \(G/F\) is finite or else \(G/F\) is infinite cyclic-by-finite, \(F\) has min-\(G\) and \(F/[F,x]\) is finite, where \(x\) is any element of infinite order in \(G\).