Centralizers of Finite p-Subgroups in Simple Locally Finite Groups

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2017-01-01
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).

Citation Formats
M. Kuzucuoğlu, “Centralizers of Finite p-Subgroups in Simple Locally Finite Groups,” JOURNAL OF SIBERIAN FEDERAL UNIVERSITY-MATHEMATICS & PHYSICS, vol. 10, no. 3, pp. 281–286, 2017, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/43775.