Maximal subgroups in finite and profinite groups

We prove that if a finitely generated profinite group is not generated with positive probability by finitely many random elements, then every finite group is obtained as a quotient of an open subgroup of . 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 has at most maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.

     

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.     

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.     

On groups whose subgroups are closed in the profinite topology

A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF.     

The minimal free resolution of a Borel ideal

We consider the Eliahou–Kervaire minimal free resolution of a 0-Borel ideal and its applications.     

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.     

On profinite groups in which commutators are covered by finitely many subgroups

For a family of group words w we show that if G is a profinite group in which all w-values are contained in a union of finitely many subgroups with a prescribed property, then the verbal subgroup w(G) has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If G contains finitely many subgroups G ₁, G ₂, . . . , G _s of finite exponent e whose union contains all γ _k -values in G, it is shown that γ _k (G) has finite (e, k, s)-bounded exponent. If G contains finitely many subgroups G ₁, G ₂, . . . , G _s of finite rank r whose union contains all γ _k -values, it is shown that γ _k (G) has finite (k, r, s)-bounded rank.     

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.

     

Pseudosymmetric equations in a free monoid

In 2011, Makanin obtained the general solution of the symmetric equation $$x_1 x_2 \cdots x_{n - 1} x_n = x_n x_{n - 1} \cdots x_2 x_1$$ in a free monoid. In the present paper, a generalization of this result is given. Namely, we shall describe the general solution of the so-called pseudosymmetric equations in a free monoid which are obtained from the symmetric equations by transposing one (any) pair of adjacent variables.     

Finite groups whose minimal subgroups are weakly H-subgroups

Let G be a finite group.A subgroup H of G is called an H-subgroup in G if NG(H) ∩Hg≤H for all g∈G.A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G=HK and H∩K is an H-subgroup in G.In this paper,we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G.Our results improve and generalize several recent results in the literature.     

The intersection of free submonoids of a free monoid is free

Several necessary and sufficient conditions exist for a submonoid of a free monoid to be free. For a few, see [1] through [6]. This short note uses one of these conditions, due to Schützenberger (see [2], p. 119, or [4], Theorem 1.4) to establish that the intersection of free submonoids of a free monoid M is again free. Schützenberger's condition is also proven in this note.     

The generation of the augmentation ideal in profinite groups

We find a formula for computing the minimum number of generators for the augmentation ideal in the profinite setting; this is a generalisation of a result obtained in the finite case by Cossey, Gruenberg and Kovács.