The structure of an infinite Sylow subgroup in some periodic Shunkov groups

V. I. Senashov

Diskretnaya Matematika, Tome 14 (2002) no. 4, pp. 133-152

Résumé

We study periodic groups such that the normaliser of any finite non-trivial subgroup of such a group is almost layer-finite. The class of groups satisfying this condition is rather wide and includes the free Burnside groups of odd period which is greater than 665 and the groups constructed by A. Yu. Olshanskii. We consider the classical question: how the properties of the system of subgroups of a group influence on the properties of the group? We show that almost layer-finiteness is transferred on the group $G$ from the normalisers of non-trivial finite subgroups of the group $G$ if $G$ is periodic conjugately biprimitively finite group with a strongly embedded subgroup. We study the structure of an infinite Sylow 2-subgroup in a periodic conjugately biprimitively finite group in the case that the normaliser of any finite non-trivial subgroup is almost layer-finite. The results of the paper can be useful in the study of the class of periodic conjugately biprimitively finite groups (periodic Shunkov groups). The research was supported by the Russian Foundation for Basic Research, grants 99–01–00432 and 02–01–00078, and by grant №9 of the Sixth Competition of Scientific Projects of Young Researchers in 1999.