TY - JOUR

T1 - Maximal subgroups in finite and profinite groups

AU - Borovik, Alexandre V.

AU - Pyber, Laszlo

AU - Shalev, Aner

PY - 1996

Y1 - 1996

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=21444458682&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-96-01665-0

DO - 10.1090/s0002-9947-96-01665-0

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AN - SCOPUS:21444458682

SN - 0002-9947

VL - 348

SP - 3745

EP - 3761

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 9

ER -