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CENTRALIZERS OF p-SUBGROUPS IN SIMPLE LOCALLY FINITE GROUPS

Part of: Linear algebraic groups and related topics Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  21 February 2019

KIVANÇ ERSOY Open the ORCID record for KIVANÇ ERSOY [Opens in a new window]
KIVANÇ ERSOY*
Affiliation:
Institute of Mathematics, Freie Universität Berlin, 14195 Berlin, Germanyand Department of Mathematics, Mimar Sinan Fine Arts University, Istanbul 34427, Turkey e-mail: ersoykivanc@gmail.com
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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.

MSC classification

Primary: 20E32: Simple groups 20E36: Automorphisms of infinite groups
Secondary: 20G15: Linear algebraic groups over arbitrary fields
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 183 - 186
Creative Commons
Creative Common License - CCCreative Common License - BY
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

References

Ersoy, K. and Gupta, C. K., Locally finite groups with centralizers of finite rank, Comm. Alg. 44(12) (2016), 5074–5087.CrossRefGoogle Scholar
Ersoy, K., Kuzucuoğlu, M. and Shumyatsky, P., Locally finite groups and their subgroups with small centralizers, J. Algebra 481 (2017), 1–11.CrossRefGoogle Scholar
Hartley, B. and Kuzucuoğlu, M., Centralizers of elements in locally finite simple groups, Proc. London Math. Soc.62 (1991), 301–324.CrossRefGoogle Scholar
Hartley, B. and Shute, G., Monomorphisms and direct limits of finite groups of Lie type, Q. J. Math. 35(1) (1984), 49–71CrossRefGoogle Scholar
Kegel, O. H. and Wehrfritz, B. A. F., Strong finiteness conditions in locally finite groups, Math. Z. 117 (1970), 309–324.CrossRefGoogle Scholar
Kegel, O. H. and Wehrfritz, B. A. F., Locally finite groups (North-Holland, Amsterdam, 1973).Google Scholar
Malle, G. and Testerman, D., Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133 (Cambridge University Press, Cambridge, 2011).CrossRefGoogle Scholar
Steinberg, R., Torsion in reductive groups, Adv. Math. 15(1) (1975), 63–92.CrossRefGoogle Scholar
Shunkov, V. P., On the minimality problem for locally finite groups, Algebra Logic 9 (1970), 137–151.CrossRefGoogle Scholar
Springer, T. A. and Steinberg, R., “Conjugacy Classes”, in Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Mathematics, vol. 131 (Springer-Verlag, Berlin, 1970).Google Scholar