Generalizations of groups in which normality is transitive

摘要

A group G is called a Hall(x)-group if G possesses a nilpotent normal subgroup N such that G/N' is an X-group. A group G is called an X-0-group if G/Phi(G) is an X-group. The aim of this article is to study finite solvable Hall(X)-groups and X-0-groups for the classes of groups T, PT, and PST. Here T, PT, and PST denote, respectively, the classes of groups in which normality, permutability, and Sylow-permutability are transitive relations. Finite solvable T-groups, PT-groups, and PST groups were globally characterized, respectively, in Gaschutz (1957), Zacher (1964), and Agrawal (1975). Here we arrive at similar characterizations for finite solvable Hall(X)-groups and X-0-groups where X is an element of{T, PT, PST}. A key result aiding in the characterization of these groups is their possession of a nilpotent residual which is a nilpotent Hall subgroup of odd order. The main result arrived at is Hall(PST) = T-0 for finite solvable groups.