Cohomology and finite subgroups of profinite groups

We prove two theorems linking the cohomology of a pro- group with the conjugacy classes of its finite subgroups.

The number of conjugacy classes of elementary abelian -subgroups of is finite if and only if the ring is finitely generated modulo nilpotent elements.

If the ring is finitely generated, then the number of conjugacy classes of finite subgroups of is finite.

     

Maximal cyclic subgroups and prime divisors in finite groups

Let G be a finite group with no chief factor simple of Lie type E ₈(q) and C a cyclic subgroup of largest order in G. It is shown that at most two primes in the open interval ([|C|/2], |C|) divide |G|.Received: 14 January 2005     

Finite index subgroups in profinite groups

We prove that every subgroup of finite index in a (topologically) finitely generated profinite group is open. This implies that the topology in such a group is uniquely determined by the group structure. The result follows from a ‘uniformity theorem’ about finite groups: given a group word w that defines a locally finite variety and a natural number d, there exists f=f_w(d) such that in every finite d-generator group G, each element of the verbal subgroup w(G) is a product of fw-values. Similar methods show that in a finite d-generator group, each element of the derived group is a product of g(d) commutators; this implies that the (abstract) derived group in any finitely generated profinite group is closed. To cite this article: N. Nikolov, D. Segal, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Commutators and pronilpotent subgroups in profinite groups

Let \(G\) be a profinite group in which all pronilpotent subgroups generated by commutators are periodic. We prove that \(G^{\prime }\) is locally finite.     

Maximal orders of Abelian subgroups in finite simple groups

We bring out upper bounds for the orders of Abelian subgroups in finite simple groups. (For alternating and classical groups, these estimates are, or are nearly, exact.) Precisely, the following result, Theorem A, is proved. Let G be a non-Abelian finite simple group and G L₂ (q), where q=p^t for some prime number p. Suppose A is an Abelian subgroup of G. Then |A|³<|G|. Our proof is based on a classification of finite simple groups. As a consequence we obtain Theorem B, which states that a non-Abelian finite simple group G can be represented as ABA, where A and B are its Abelian subgroups, iff G≌ L₂(2^t) for some t ≥ 2; moreover, |A|-2^t+1, |B|=2^t, and A is cyclic and B an elementary 2-group. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 131–160, March–April, 1999.     

Normal subgroups of profinite groups of non-negative deficiency

The principal goal of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group G of non-negative deficiency gives rather strong consequences for the structure of G. To make this precise we introduce the notion of p-deficiency (p a prime) for a profinite group G. We prove that if the p-deficiency of G is positive and N is a finitely generated normal subgroup such that the p  -Sylow subgroup of G/N$G/N$ is infinite and p divides the order of N   then we have _cdp(G)=2${\mathrm{cd}}_p(G)=2$, _cdp(N)=1${\mathrm{cd}}_p(N)=1$ and _vcdp(G/N)=1${\mathrm{vcd}}_p(G/N)=1$ for the cohomological p-dimensions; moreover either the p  -Sylow subgroup of G/N$G/N$ is virtually cyclic or the p-Sylow subgroup of N is cyclic. If G is a profinite Poincaré duality group of dimension 3 at a prime p   (PD³${\mathit{PD}}^3$-group at p) we show that for N and p as above either N   is PD¹${\mathit{PD}}^1$ at p   and G/N$G/N$ is virtually PD²${\mathit{PD}}^2$ at p or N   is PD²${\mathit{PD}}^2$ at p   and G/N$G/N$ is virtually PD¹${\mathit{PD}}^1$ at p.     

Maximal subgroups of exceptional groups of Lie type,finite and algebraic

The maximal closed subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields are determined, subject to certain restrictions on the characteristic. This result is used to prove a reduction theorem for maximal subgroups of finite exceptional groups of Lie type: such a subgroup is either of known type, or it is almost simple. Finally, the generic almost simple maximal subgroups are determined for large characteristic.Dedicated to Professor J. Tits on the occasion of his sixtieth birthdaySupported in part by NSF and NSA grants and an SERC visiting fellowship at Imperial College, London.     

Non-free torsion-free profinite groups with open free subgroups

For every integere≧3 there exists a non-free torsion-free profinite group containing a as an open subgroup. Supported by Rothschild Fellowship.     

Prosolvable subgroups of free products of profinite groups

In this paper we establish several equivalent conditions for a commutative ring in which every principal ideal is a finite intersection of prime power ideals to be a general ZPI-ring. Using these results, we establish some equivalent conditions for a commutative ring in which every principal ideal is a finite intersection of primary ideals to be a general ZPI-ring.     

On profinite groups with finite abelianizations

Profinite groups with finite p-abelianizations arise in various contexts: group theory, number theory and geometry. Using Ph. Furtw?ngler’s transfer vanishing theorem it will be proved that a finitely generated profinite group Ĝ with this property satisfies 〚Ĝ〛) = 0 (Thm. A). As a consequence one finds that a hereditarily just-infinite non-virtually cyclic pro-p group has only one end (Cor. B). Applied to 3-dimensional Poincaré duality groups, Theorem A yields a generalization of A. Reznikov’s theorem on 3-dimensional co-compact hyperbolic lattices violating W. Thurston’s conjecture (Thm. C).     

Finitely generated subgroups of free profinite groups and of some Galois groups

It is proved that, for a (closed) subgroupH of a free profinite or free prosolvable groupF of rankF>1 such thatH contains a nontrivial composition subgroupN ofF, we have rankF< and [F:H]<.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 95–106, July, 1998.