Inert subgroups in infinite simple groups

INERT SUBGROUPS IN INFINITE SIMPLE GROUPS( M.Sc. Thesis )Tuğba USTAGAZ UNIVERSITYINSTITUTE OF SCIENCE AND TECHNOLOGYMay, 2006ABSTRACTIn the present thesis, we consider inert subgroupsof a group G. In fact, inertsubgroups appear in various section of group theory arouses a natural desire tomaket hem a subject of especial research. The term ?inert subgroup? was coined byKegel.A subgroup H of a group G is called an inert subgroup of G, if [H : H ∩ H g ] < ∞ forall g ∈ G . It is clear that every normal subgroup and every finite subgroup areinert. Consequently, the notion of inert subgroup generalizes the notion of normalsubgroup.The present study consists of seven chapters. The second chapter includes definitionsand theorems which we use in further chapters. In the third chapter, we obtain basicproperties and examples of inert subgroups. In the fourth chapter, we discuss thecommensurable property and some observations. In the fifth chapter, we investigatethe centralizer of involutions in alternating groups and simple groups of Lie type.Finally, in the sixth chapter, we discuss the FC-efect of inert subgroups, radicals ininert subgroups and inert subgroups in simple groups in detail.Science Code : 204.1.025Key Words : Inert subgroups, involution, locally finite groups, commensurablePage Number : 87Adviser : Assist. Prof. Dr. Aynur ARIKAN