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Centralizers of abelian subgroups in locally finite simple groups

Published online by Cambridge University Press:  20 January 2009

M. Kuzucuoǧlu
M. Kuzucuoǧlu
Affiliation:
Department of Mathematics, Middle East Technical University, 06531, Ankara, Turkey E-mail address: matmah@rorqual.cc.metu.edu.tr
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Abstract

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It is shown that, if a non-linear locally finite simple group is a union of finite simple groups, then the centralizer of every element of odd order has a series of finite length with factors which are either locally solvable or non-abelian simple. Moreover, at least one of the factors is non-linear simple. This is also extended to abelian subgroup of odd orders.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 2 , June 1997 , pp. 217 - 225
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

1. Belyaev, V. V., Locally finite Chevalley groups, Studies in Group Theory (Acad. of Sciences of the U.S.S.R. Urals Scientific Centre, 1984).Google Scholar
2. Borovik, A. V., Embeddings of finite Chevalley groups and periodic linear groups, Sibirsky. Mat. Zh. 24 (1983), 26–35.Google Scholar
3. Carter, R. W., Simple groups of Lie Type (John Wiley London, 1972).Google Scholar
4. Carter, R. W., Finite groups of Lie Type (John Wiley London, 1985).Google Scholar
5. Hartley, B., On some simple locally finite groups constructed by Meierfrankenfeld, J. London Math. Soc. (2) 52 (1995), 345–355.CrossRefGoogle Scholar
6. Hartley, B., Centralizing properties in simple locally finite groups and large finite classical groups, J. Austral. Math. Soc. Ser. A 49 (1990), 502–513.CrossRefGoogle Scholar
7. Hartley, B. and Shute, G., Monomorphisms and direct limits of finite groups of Lie type, Quart. J. Math. Oxford(2) 35 (1984), 49–71.CrossRefGoogle Scholar
8. Hartley, B. and Kuzucuoğlu, M., Centralizers of elements in locally finite simple groups, Proc. London Math. Soc. (3) 62 (1991), 301–324.CrossRefGoogle Scholar
9. Kegel, O. H. and Wehrfritz, B., Locally Finite Groups (North Holland, Amsterdam, 1973).Google Scholar
10. Kuzucuoğlu, M., Centralizers of semisimple subgroups in locally finite simple groups, Rend. Sem. Mat. Univ. Padova 92 (1994), 79–90.Google Scholar
11. Kuzucuoğlu, M., Barely Transitive Permutation Groups, Arch. Math. 55 (1990), 521–532.CrossRefGoogle Scholar
12. Springer, T. A. and Steinberg, R., Conjugacy Classes, in Seminar on Algebraic Groups and Related Finite Groups (Lecture Notes in Math. Vol. 131, Springer Verlag, Berlin Heidelberg New York, 1970).Google Scholar
13. Springer, T. A., Proceedings Symposia Pure Mathematics Vol. 29 (Providence R.I. American Mathematical Society, 1975), 373–391.Google Scholar
14. Steinberg, R., Endomorphism of Algebraic Groups (Mem. Amer. Math. Soc. No. 80, American Math. Soc. Providence, R.I., 1968).Google Scholar
15. Thomas, S., The classification of the simple periodic linear groups, Arch. Math. 41 (1983), 103–116.CrossRefGoogle Scholar