On generalizations of finite groups in which normality is a transitive relation.

摘要

A group G is called a Hallchi-group if G possesses a nilpotent normal subgroup N such that G/N' is an chi-group. A group G is called an chio-group if G/phi(G) is an chi-group. The aim of this work is to study finite solvable Hallchi-groups and chio-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 [13], [29], and [1] whereas local characterizations were given, respectively, in [23], [6], and [4]. Here we arrive at similar global and local characterizations of finite solvable Hallchi-groups and chio-groups where chi ∈ { T , PT , PST }. Chapter one is concerned with background information on the classes T , PT , and PST . In chapter two, both the global and local characterization theorems for Hallchi and chio are given. A key result aiding in the characterization of these groups is that they possess a nilpotent residual which is nilpotent and a Hall subgroup of odd order. The main result shown here is that HallPST=To for finite solvable groups. Chapter three discusses minimal-non-chi-groups and the concept of subgroup-closure for the classes of groups under consideration. One main result shown here is that finite subgroup-closed HallPST -groups and finite solvable PST -groups are one and the same. The other main result is that finite minimal-non- HallPST -groups are precisely the minimal-non- PST -groups.