Локально ступінчасті 𝑝-групи із скінченною NSC-підгрупою

Abstract

Стаття присвячена вивченню груп G, в яких кожна підгрупа, що містить деяку фіксовану підгрупу (NSC-підгрупу), має нормальне доповнення в G. Встановлено структуру локально-ступінчастих 𝑝-груп із власною скінченною NSC-підгрупою.One of the main directions of investigations in abstract group theory is a study of groups in which some subgroups or systems of subgroups satisfy particular restricting requirements, e.g. are complemented. It was natural to consider groups G in which every subgroup containing some fixed subgroup X is complemented. Such subgroup X has been called supercomplemented in G. It turned out that the class of groups with a finite supercomplemented subgroup is extremely wide. Therefore it was decided to apply certain restrictions which allowed rather detailed description of subclasses obtained. This paper is devoted to the study of groups G in which every subgroup containing some fixed subgroup A has a normal complement in G. Such subgroup is called an NSC-subgroup. A structure of locally graded p-groups with a proper finite NSC-subgroup is established (Theorem 1). At the same time it is shown that the requirement for a group G to be locally graded is essential (Theorem 2).