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Maximal subgroups in finite and profinite groups
Author(s):
Alexandre
V.
Borovik;
Laszlo
Pyber;
Aner
Shalev
Journal:
Trans. Amer. Math. Soc.
348
(1996),
3745-3761.
MSC (1991):
Primary 20E28, 20D99;
Secondary 20B35, 20D06
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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), 203--220, 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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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 H-1364, Hungary
Email:
H1130Pyb@HUELLA.EARN
Aner
Shalev
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Email:
shalev@math.huji.ac.il
DOI:
10.1090/S0002-9947-96-01665-0
PII:
S 0002-9947(96)01665-0
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.
Copyright of article:
Copyright
1996,
American Mathematical Society
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