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Inert subgroups and centralizers of involutions in locally finite simple groups

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2003
Özyurt, Erdal
A subgroup H of a group G is called inert if [H : H \ Hg] is finite for all g 2 G. A group is called totally inert if every subgroup is inert. Among the basic properties of inert subgroups, we prove the following. Let M be a maximal subgroup of a locally finite group G. If M is inert and abelian, then G is soluble with derived length at most 3. In particular, the given properties impose a strong restriction on the derived length of G. We also prove that, if the centralizer of every involution is inert in an infinite locally finite simple group G, then every finite set of elements of G can not be contained in a finite simple group. In a special case, this generalizes a Theorem of Belyaev-Kuzucuoæglu-Seğckin, which proves that there exists no infinite locally finite totally inert simple group.