Abstract
Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite -semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence {(Gi , 1) : i } consisting of finite simple subgroups. Then for any finite subgroup F consisting of -semisimple elements in G, the centralizer CG (F) has an infinite abelian subgroup A isomorphic to a direct product of pi for infinitely many distinct primes pi .
Moreover we prove that if G is a non-linear simple locally finite group which has a Kegel sequence {(Gi , 1) : i } consisting of finite simple subgroups Gi and F is a finite -semisimple subgroup of G, then CG (F) involves an infinite simple non-linear locally finite group provided that the finite fields ki over which the simple group Gi is defined are splitting fields for Li , the inverse image of F in i for all i . The group i is the inverse image of Gi in the corresponding universal central extension group.
Comments (0)