Centralizers of Finite p-Subgroups in Simple Locally Finite Groups

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JOURNAL OF SIBERIAN FEDERAL UNIVERSITY-MATHEMATICS & PHYSICS, vol.10, no.3, pp.281-286, 2017 (ESCI) identifier identifier


We are interested in the following questions of B. Hartley: (1) Is it true that, in an infinite, simple locally finite group, if the centralizer of a finite subgroup is linear, then G is linear? (2) For a finite subgroup F of a non-linear simple locally finite group is the order vertical bar CG(F)vertical bar infinite? We prove the following: Let G be a non-linear simple locally finite group which has a Kegel sequence K = {(G(i), 1) : i is an element of N} consisting of finite simple subgroups. Let p be a fixed prime and s is an element of N. Then for any finite p subgroup F of G, the centralizer C-G(F) contains subgroups isomorphic to the homomorphic images of SL(s, F-q). In particular C-G(F) is a non-linear group. We also show that if F is a finite p-subgroup of the infinite locally finite simple group G of classical type and given s is an element of N and the rank of G is sufficiently large with respect to vertical bar F vertical bar and s, then C-G(F) contains subgroups which are isomorphic to homomorphic images of SL(s, K).