Maximal subgroups in finite and profinite groups

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 IGIc maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.

[1]  Andrea Lucchini,et al.  Generation of Almost Simple Groups , 1995 .

[2]  R. Guralnick,et al.  Solvable generation of groups and Sylow subgroups of the lower central series , 1982 .

[3]  Cheryl E. Praeger,et al.  On the orders of Primitive Permutation Groups , 1980 .

[4]  Cheryl E. Praeger,et al.  On the O'Nan-Scott theorem for finite primitive permutation groups , 1988, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[5]  Martin W. Liebeck,et al.  The Subgroup Structure of the Finite Classical Groups , 1990 .

[6]  R. Carter.  THE SUBGROUP STRUCTURE OF THE FINITE CLASSICAL GROUPS (London Mathematical Society Lecture Note Series 129) , 1991 .

[7]  William M. Kantor,et al.  The probability of generating a finite classical group , 1990 .

[8]  Avinoam Mann,et al.  Positively finitely generated groups , 1996 .

[9]  William M. Kantor,et al.  The probability of generating a classical group , 1994 .

[10]  A. Lubotzky.  Subgroup growth and congruence subgroups , 1995 .

[11]  A. Shalev.  Growth Functions, p‐Adic Analytic Groups, and Groups of Finite Coclass , 1992 .

[12]  László Pyber,et al.  Groups with super-exponential subgroup growth , 1996, Comb..

[13]  Avinoam Mann,et al.  Finitely generated groups of polynomial subgroup growth , 1993 .

[14]  I. G. MacDonald,et al.  Lectures on Lie Groups and Lie Algebras: Simple groups of Lie type , 1995 .

[15]  A. Shalev,et al.  Simple groups, maximal subgroups, and probabilistic aspects of profinite groups , 1996 .

[16]  Michael Aschbacher,et al.  On the maximal subgroups of the finite classical groups , 1984 .

[17]  László Pyber,et al.  Enumerating finite groups of given order , 1993 .

[18]  Peter J. Cameron,et al.  On the orders of primitive groups with restricted nonabelian composition factors , 1982 .

[19]  The probability of generating certain profinite groups by two elements , 1994 .