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 -