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Maximal subgroups in finite and profinite groups |
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| Authors: | Alexandre V. Borovik Laszlo Pyber Aner Shalev |
| Affiliation: | Department of Mathematics, University of Manchester, Institute of Science and Technology, P.O. Box 88, Manchester M60 1QD, United Kingdom ; Mathematical Institute, Hungarian Academy of Science, P.O.B. 127, Budapest H-1364, Hungary ; Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel |
| 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 thenprove 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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