Pronormality and permutability in periodic FC-groups
N. A. Turbay · Matematychni Studii · 2008
Pronormality and permutability in periodic FCgroups, Matematychni Studii, 29 (2008) 132138.In this paper we study periodic locally soluble F Cgroups in which permutability is a transitive relation and the class of groups in which all nite normal subgroups of Sylow subgroups are pronormal.The choice of class of periodic F C-groups is made, because it has suciently developed structure and is similar to the class of nite groups.Í. À. Òóðáàé.Ñâîéñòâà ïðîíîðìàëüíîñòè è ïåðåñòàâëÿåìîñòè â ïåðèîäè÷åñêèõ FC ãðóïïàõ // Ìàòåìàòè÷íi Ñòóäi¨.2008.Ò.29, 2. C.132138.Pàññìàòðèâàþòñÿ ïåðèîäè÷åñêèå ëîêàëüíî ðàçðåøèìûå F Cãðóïïû, â êîòîðûõ ñâîéñòâî ïåðåñòàâëÿåìîñòè ÿâëÿåòñÿ òðàíçèòèâíûì, è êëàññ ãðóïï, â êîòîðûõ âñå êîíå÷íûå íîðìàëüíûå ïîäãðóïïû ñèëîâñêèõ ïîäãðóïï ïðîíîðìàëüíû.Âûáîð êëàññà ïåðèîäè-÷åñêèõ F Cãðóïï îáóñëîâëåí òåì îáñòîÿòåëüñòâîì, ÷òî îí èìååò äîñòàòî÷íî ðàçâèòóþ ñèëîâñêóþ ñòðóêòóðó è ïî ìíîãèì àñïåêòàì áëèçîê ê êëàññó êîíå÷íûõ ãðóïï.The study of the some basic properties of the permutable subgroups started rather long time ago (see, for example [4]).In particular, there were described the groups (nite and innite) in which every subgroup is permutable [4, Theorem 2.4].Like normality, the relation to be permutable subgroup is not transitive.A nite group in which permutability is a transitive relation is said to be a P T -group.Finite soluble P T -groups have been described by G. Zacher [6].The structure of arbitrary nite P T -groups has been considered by D.J.S. Robinson [3].The study of innite P T -groups is not started yet.In paper [7], [8] we have been started to consider periodic locally soluble F C-groups in which permutability is a transitive relation.The choice of the class of periodic F C-groups is made, because it has suciently developed Sylow structure, and many another its properties are similar to those of nite groups.Therefore it is natural to begin consideration of innite P T -groups and their characterization in this class.In this paper we consider the relationship between properties of permutability and pronormality of innite periodic F C-groups.In particular, we extend the main result of the paper [1, Theorem A and Theorem D] to the class of innite periodic locally soluble F Cgroups.