Maximal subgroups in finite and profinite groups

Authors:
Alexandre V. Borovik, Laszlo Pyber and Aner Shalev

Journal:
Trans. Amer. Math. Soc. **348** (1996), 3745-3761

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), 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:
https://doi.org/10.1090/S0002-9947-96-01665-0

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.

Article copyright:
© Copyright 1996
American Mathematical Society