Maximal subgroups in finite and profinite groups

Alexandre V. Borovik*, Laszlo Pyber, Aner Shalev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. 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 G has at most |G|c maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.

Original languageAmerican English
Pages (from-to)3745-3761
Number of pages17
JournalTransactions of the American Mathematical Society
Issue number9
StatePublished - 1996


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