×

Finite groups have local non-Schur centralizers. (English) Zbl 0820.20025

Using the classification of the finite simple groups the authors prove the following theorem: If \(G\) is a finite group of order divisible by the prime \(p\) then \(G\) contains a \(p\)-singular element \(g\) whose \(p\)-part is not contained in the commutator subgroup of \(C_ G(g)\).

MSC:

20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D25 Special subgroups (Frattini, Fitting, etc.)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Conway, J., ATLAS of finite groups (1985), Oxford: Clarendon Press, Oxford · Zbl 0568.20001
[2] M. Aschbacher, Finite Group theory,Cambridge University Press (1988) · Zbl 0826.20001
[3] Chang, B., The conjugate classes of Chevalley groups of type (G_2)*, J. of Algebra, 9, 190-211 (1968) · Zbl 0285.20043
[4] Deriziotis, D.; Michler, G. O., Character tables and blocks of finite simple triality groups^3D_4(q), Trans. Amer. Math Soc., 303, 39-70 (1987) · Zbl 0628.20014
[5] Enomoto, H., The conjugate classes of Chevalley groups of typeG_2 over finite fields of characteristic 2 or 3, J. Fac. Sci. Univ. Tokyo, 16, 497-512 (1969) · Zbl 0242.20049
[6] Fleischmann, P.; Janiszczak, I., The semisimple conjugacy classes of finite groups of Lie type E_6 and E_7, Comm. in Alg., 21, 93-161 (1993) · Zbl 0813.20015
[7] P. Fleischmann, I. Janiszczak, The semisimple conjugacy classes and the generic class numbers of Chevalley groups of typeE_s, Preprint No. 17, Inst. f. Exp. Math., (1992)
[8] M. Gerstenhaber, D.J. Green, A group theoretic consequence of the Donald - Flanigan conjecture, Preprint No. 14, Inst. f. Exp. Math.,(1992) · Zbl 0805.16030
[9] Gorenstein, D., Finite Groups (1968), New York: Harper and Row, New York · Zbl 0185.05701
[10] Huppert, B., Endliche Gruppen I (1967), Berlin: Springer, Berlin · Zbl 0217.07201
[11] Huppert, B.; Blackburn, N., Finite Groups III (1982), Berlin: Springer, Berlin · Zbl 0514.20002
[12] Shinoda, K., The conjugacy classes of Chevalley groups of typeF_4 over finite fields of characteristic 2, J. Fac. Sci. Univ. Tokyo, 21, 133-159 (1974) · Zbl 0306.20013
[13] Shinoda, K., The conjugacy classes of the finite Ree groups of typeF_4, J. Fac. Sci. Univ. Tokyo, 22, 1-15 (1975) · Zbl 0306.20014
[14] Shoji, T., The conjugacy classes of Chevalley groups of typeF_4 over finite fields of characteristic p # 2, J. Fac. Sci. Univ. Tokyo, 21, 1-17 (1974) · Zbl 0279.20038
[15] T. A. Springer, R. Steinberg, „Conjugacy Classes“, in „Seminar on algebraic groups and related topics”, ed. Borel et al.,Springer LNM 131,(1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.