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CENTRALIZERS OF p-SUBGROUPS IN SIMPLE LOCALLY FINITE GROUPS
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Linear algebraic groups and related topics
Structure and classification of infinite or finite groups
Published online by Cambridge University Press: 21 February 2019
Abstract
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In Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra481 (2017), 1–11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikov and for every non-identity α ∈ A the set of fixed points CG(α) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) is Chernikov and for every non-identity α ∈ P the set of fixed points CG(α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ≅PSLp(k) where char k ≠ p and P has a subgroup Q of order p2 such that CG(P) = Q.
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.- Copyright
- Copyright © Glasgow Mathematical Journal Trust 2019
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