Soluble groups with extremal conditions on commutators. (Q996951)

This paper investigates two classes of soluble groups, namely, \(\mathfrak X\)-groups (\(\mathfrak N\)-groups), defined by the property that the commutator of any two normal subgroups coincides with their intersection (is trivial). These notions come from the theory of \textit{Y. Kurata} [Osaka J. Math. 1, 201-229 (1964; Zbl 0129.01601)] on prime and primary subgroups. After establishing basic properties of \(\mathfrak X\)-groups, such as that \(\mathfrak X\)-groups are necessarily finite, the author characterizes these groups: a solvable group is an \(\mathfrak X\)-group if and only if the factor \(G/G'\) is also an \(\mathfrak X\)-group and every normal subgroup either contains the commutator subgroup or is a term of the Fitting series. \(\mathfrak X\)-groups have a lattice-theoretical property as well. A \(K\)-group is a group with complemented subgroup lattice. In fact, an \(\mathfrak X\)-group is a \(K\)-group if and only if the factor \(G/G'\) is not cyclic of prime square order. Concerning \(\mathfrak N\)-groups, it is proved that a solvable group is an \(\mathfrak N\)-group if and only if the Fitting subgroup is Abelian and every proper normal subgroup is contained in the Fitting subgroup.