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On extensions of the Baer-Suzuki theorem. (English) Zbl 0794.20029
Using the classification of the finite simple groups the authors prove the following theorem which is the main result of the paper: Let $p$ be a prime and $G$ be a finite group containing an element $x\in G$ of order $p$ such that $[x,g]$ is a $p$-element for every $g\in G$. Then $x\in O\sb p(G)$. The authors point out that this theorem was obtained also by {\it W. Xiao} [Sci. China, Ser. A 34, No. 9, 1025-1031 (1991; Zbl 0743.20015)] for a $p$-element of arbitrary order.

20D20Sylow subgroups of finite groups, Sylow properties, $\pi$-groups, $\pi$-structure
20E07Subgroup theorems; subgroup growth
20F45Engel conditions on groups
20F12Commutator calculus (group theory)
Full Text: DOI
[1] J. Alperin and R. Lyons,Conjugacy classes of p-elements, J. Algebra19 (1971), 536--537. · Zbl 0238.20026 · doi:10.1016/0021-8693(71)90086-X
[2] O. D. Artemovich,Isolated elements of prime order in finite groups, Ukranian Math. J.40 (1988), 397--400. · Zbl 0662.20016
[3] M. Aschbacher,The 27-dimensional module for E 6,IV, J. Algebra131 (1990), 23--39. · Zbl 0698.20031 · doi:10.1016/0021-8693(90)90164-J
[4] M. Aschbacher,Overgroups of Sylow subgroups in sporadic groups, Memoirs of the Amer. Math. Soc.60 (1986), No. 343. · Zbl 0585.20005
[5] A. Borel,Linear Algebraic Groups, 2nd Ed., Springer-Verlag, New York, 1991. · Zbl 0726.20030
[6] W. Feit,The Representation Theory of Finite Groups, North-Holland Publishing Company, Amsterdam, 1982. · Zbl 0493.20007
[7] D. Gorenstein,Finite Groups, Harper & Row, New York, 1968. · Zbl 0185.05701
[8] D. Gorenstein,Finite Simple Groups -- An Introduction to their Classification, Plenum Press, New York, 1982. · Zbl 0483.20008
[9] F. Gross,Automorphisms which centralize a Sylow p-subgroup, J. Algebra77 (1982), 202--233. · Zbl 0489.20019 · doi:10.1016/0021-8693(82)90287-3
[10] J. Humphreys,Linear Algebric Groups, Springer-Verlag, New York, 1975. · Zbl 0325.20039
[11] G. Seitz,Generation of finite groups of Lie type, Trans. Amer. Math. Soc.271 (1982), 351--407. · Zbl 0514.20013 · doi:10.1090/S0002-9947-1982-0654839-1
[12] R. Steinberg,Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc.,80, 1968. · Zbl 0164.02902
[13] H. N. Ward,On Ree’s series of simple groups, Trans. Amer. Math. Soc.121 (1966), 62--89. · Zbl 0139.24902
[14] H. Wielandt,Kriterien für Subnormalität in endlichen Gruppen, Math Z.138 (1974), 199--203. · Zbl 0275.20041 · doi:10.1007/BF01237117
[15] Wen-Jun Xiao,Glauberman’s conjecture, Mazurov’s problem and Peng’s problem, Science in China Series A34 (1991), 1025--1031. · Zbl 0743.20015