Hall Subgroups of Odd Order in Finite Groups
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We complete the description of Hall subgroups of odd order in finite simple groups initiated by F. Gross, and as a consequence, bring to a close the study of odd order Hall subgroups in all finite groups modulo classification of finite simple groups. In addition, it is proved that for every set π of primes, an extension of an arbitrary D π -group by a D π-group is again a D π -group. This result gives a partial answer to Question 3.62 posed by L. A. Shemetkov in the “Kourovka Notebook.”.
- A. S. Kondratiev, "Subgroups of finite Chevalley groups," tiUsp. Mat. Nauk, 41, No. 1(247), 57-96 (1986).
- tiThe Kourovka Notebook, Institute of Mathematics SO RAN, Novosibirsk (1999).
- Ph. Hall, "Theorems like Sylow's," tiProc. London Math. Soc., III. Ser., 6, No. 22, 286-304 (1956).
- J. G. Thompson, "Hall subgroups of the symmetric groups," J. Comb. Theory, Ser. A, No. 1, 271-279 (1966).
- F. Gross, "On a conjecture of Philip Hall," Proc. London Math. Soc., III. Ser., 52, No. 3, 464-494 (1986).
- F. Gross, "Hall subgroups of order not divisible by 3," Rocky Mountain J. Math., 23, No. 2, 569-591 (1993).
- D. O. Revin, "Hall π-subgroups of finite Chevalley groups whose characteristic belongs to π," Mat. Trudy, 2, No. 1, 160-208 (1999).
- D. O. Revin, "Two D π-theorems for a class of finite groups," preprint No. 40, NIIDM, Novosibirsk (1999).
- F. Gross, "Odd order Hall subgroups of the classical linear groups," Math. Z. 220, No. 3, 317-336 (1995).
- F. Gross, "Conjugacy of odd order Hall subgroups," Bull. London Math. Soc. 19, No. 4, 311-319 (1987).
- M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1996).
- R. W. Carter, Simple Groups of Lie Type, Pure Appl. Math., Vol. 28, Wiley, London (1972).
- J. E. Humphreys, Linear Algebraic Groups, Springer, New York (1975).
- R. W. Carter, "Centralizers of semisimple elements in finite groups of Lie type," Proc. London Math. Soc., III. Ser. 37, No. 3, 491-507 (1978).
- R. W. Carter, "Conjugacy classes in the Weyl group," Comp. Math. 25, No. 1, 1-59 (1972).
- J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Math. Surv. Mon., Vol. 43, Am. Math. Soc., Providence, R.I. (1995).
- D. Deriziotis, "Conjugacy classes and centralizers of semisimple elements in finite groups of Lie type," Vorlesungen aus dem Fachbereich Mathetmatic der Universität Essen, Heft 11 (1984).
- R. W. Carter, "Centralizers of semisimple elements in finite classical groups," Proc. London Math. Soc., III. Ser. 42, No. 1, 1-41 (1981).
- D. Deriziotis, "The centralizers of semisimple elements of the Chevalley groups E 7 and E 8," Tokyo J. Math. 6, No. 1, 191-216 (1983).
- M. Suzuki, "On a class of doubly transitive groups," Ann. Math., II. Ser. 75, No. 1, 105-145 (1962).
- H. N. Ward, "On Ree's series of simple groups," Trans. Am. Math. Soc. 121, No. 1, 62-80 (1966).
- A. V. Borovik, "The structure of finite subgroups of simple algebraic groups," Algebra Logika 28, No. 3, 249-279 (1989).
- V. D. Mazurov and D. O. Revin, "The Hall D π-property for finite groups," Sib. Mat. Zh. 38, No. 1, 125-134 (1997).
- R. Steinberg, Lectures on Chevalley Groups, Yale University (1967).
- A. V. Borovik, "Jordan subgroups of simple algebraic groups," Algebra Logika, 28, No. 2, 144-159 (1989).
- D. Gorenstein and R. Lyons, The Local Structure of Finite Groups of Characteristic 2 Type, Mem. Am. Math. Soc. Vol. 42(276), Providence, R.I. (1983).
- Hall Subgroups of Odd Order in Finite Groups
Algebra and Logic
Volume 41, Issue 1 , pp 8-29
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- finite simple group, Hall subgroup, exceptional groups of Lie type
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