Maximal subgroups in finite and profinite groups
Authors:
Alexandre V. Borovik, Laszlo Pyber and Aner Shalev
Journal:
Trans. Amer. Math. Soc. 348 (1996), 37453761
MSC (1991):
Primary 20E28, 20D99; Secondary 20B35, 20D06
MathSciNet review:
1360222
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Abstract: We prove that if a finitely generated profinite group is not generated with positive probability by finitely many random elements, then every finite group is obtained as a quotient of an open subgroup of . The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203220, we then prove that a finite group has at most maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.
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 M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469514. MR 86a:20054
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 M. Aschbacher and R. Guralnick, Solvable generation of groups and Sylow subgroups of the lower central series, J. Algebra 77 (1982), 189201. MR 84c:20025
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 L. Babai, P. J. Cameron, P. P. Pálfy, On the orders of primitive permutation groups with restricted nonabelian composition factors, J. Algebra 79 (1982), 161168. MR 84e:20003
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 M. Bhattacharjee, The probability of generating certain profinite groups by two elements, Israel J. Math. 86 (1994), 311329. MR 95c:20039
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 F. Dalla Volta and A. Lucchini, Generation of almost simple groups, Preprint.
 7.
 W. Kantor and A. Lubotzky, The probability of generating a finite classical group, Geom. Ded. 36 (1990), 6787. MR 91j:20041
 8.
 P. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser. 129, Cambridge University Press, Cambridge, 1990. MR 91g:20001
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 V. Landazuri and G. M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418443. MR 50:13299
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 A. Mann, Positively finitely generated groups, Forum Math., to appear.
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 L. Pyber, Enumerating finite groups of given order, Ann. of Math. 137 (1993), 203220. MR 93m:11097
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Additional Information
Alexandre V. Borovik
Affiliation:
Department of Mathematics, University of Manchester, Institute of Science and Technology, P.O. Box 88, Manchester M60 1QD, United Kingdom
Email:
borovik@lanczos.ma.umist.ac.uk
Laszlo Pyber
Affiliation:
Mathematical Institute, Hungarian Academy of Science, P.O.B. 127, Budapest H1364, Hungary
Email:
H1130Pyb@HUELLA.EARN
Aner Shalev
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Email:
shalev@math.huji.ac.il
DOI:
http://dx.doi.org/10.1090/S0002994796016650
PII:
S 00029947(96)016650
Received by editor(s):
September 21, 1995
Additional Notes:
The second author acknowledges support of the Hungarian National Foundation for Scientific Research, Grant No.\ T7441.
The third author acknowledges support of the Basic Research Foundation, administrated by the Israel Academy of Sciences and Humanities.
Article copyright:
© Copyright 1996
American Mathematical Society
