Maximal subgroups in finite and profinite groups

Abstract

Abstract. 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.

Random generation of finite and profinite groups and group enumeration

“…Perhaps the most revealing is (3). The equivalence PFG ⇔ (4) extends the main result of [8] which states that finitely generated non-PFG groups have arbitrarily large alternating sections. The equivalence PFG ⇔ (7) also follows from Theorem 1.…”

“…For primitive soluble groups it is an immediate consequence of [47,Lemma 3.4]. More generally in [8] it was shown to hold for groups G with no large alternating sections (in which case the primitive groups have polynomial size [5]). Moreover by a central result of [38], it was known to hold for primitive groups with a given abstract isomorphism type.…”

“…M is non-abelian. Then M ∼ = S s = S 1 × · · · × S s for some non-abelian simple group S and some s. By [8,Lemma 2.3] there are at most O(n) possibilities for S. Hence there are at most O(n 2 ) possibilities for M . We fix one of them.…”

“…On the other hand, Mann [35] has proved that finitely generated prosoluble groups have this property. More generally in [8] it was shown that finitely generated profinite groups which do not have arbitrarily large alternating sections are PFG.…”

Loops of Lie type: a p-adic example

“…[57] page 67. 14 Cfr. theorem 3.5.1 page 60 and corollary 3.5.2. page 62 in [53], the more general case only requires the condition p > 2.…”