×

zbMATH — the first resource for mathematics

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.)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] J. Conway et al., ATLAS of finite groups, Clarendon Press, Oxford (1985) · Zbl 0568.20001
[2] M. Aschbacher, Finite Group theory,Cambridge University Press (1988) · Zbl 0583.20001
[3] B. Chang, The conjugate classes of Chevalley groups of type (G 2)*,J. of Algebra,9(1968), 190–211 · Zbl 0285.20043
[4] D. Deriziotis, G.O. Michler, Character tables and blocks of finite simple triality groups3D4(q),Trans. Amer. Math Soc.,303(1987), 39–70 · Zbl 0628.20014
[5] H. Enomoto, The conjugate classes of Chevalley groups of typeG 2 over finite fields of characteristic 2 or 3,J. Fac. Sci. Univ. Tokyo,16(1969, 497–512
[6] P. Fleischmann, I. Janiszczak, The semisimple conjugacy classes of finite groups of Lie type E6 and E7,Comm. in Alg.,21(1993), 93–161 · 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] D. Gorenstein, Finite Groups, Harper and Row, New York (1968)
[10] B. Huppert, Endliche Gruppen I, Springer, Berlin, (1967) · Zbl 0217.07201
[11] B. Huppert, N. Blackburn, Finite Groups III, Springer, Berlin, (1982) · Zbl 0514.20002
[12] K. Shinoda, The conjugacy classes of Chevalley groups of typeF 4 over finite fields of characteristic 2,J. Fac. Sci. Univ. Tokyo,21(1974), 133–159 · Zbl 0306.20013
[13] K. Shinoda, The conjugacy classes of the finite Ree groups of typeF 4,J. Fac. Sci. Univ. Tokyo,22(1975), 1–15 · Zbl 0306.20014
[14] T. Shoji, The conjugacy classes of Chevalley groups of typeF 4 over finite fields of characteristic p # 2,J. Fac. Sci. Univ. Tokyo,21(1974), 1–17 · 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.