ON THE STRUCTURE OF THE NORMAL-SUBGROUPS OF A GROUP - NILPOTENCY

A group G is said to have property-nu if, whenever N is a normal non-nilpotent subgroup of G, there is a finite non-nilpotent G-quotient of N. Polycyclic-by-finite groups, free groups, subgroups of GL (n, Z), subgroups of finitely generated abelian-by-nilpotent-by-finite groups, and free metanilpotent groups satisfy property-nu. Fit (G) is the Fitting subgroup of G, phi (G) is the Frattini-subgroup of G, and phi(f) (G) is the intersection of all maximal subgroups of G of finite index in G (here phi(f) (G) = G if no maximal subgroups of finite index in G exist). A group G has property v if and only if phi(f) (G) and Fit (G) are nilpotent subgroups of G and Fit (G/phi(f), (G)) = Fit (G)/phi(f) (G). A group G of finite rank has property-nu if and only if G is soluble-by-finite, phi (G) and Fit (G) are nilpotent subgroups of G, and Fit (G/phi(G)) = Fit (G)/phi(G).