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.     

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.     

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.     

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.     

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.     

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ? D₈. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p ⁴, (h) a maximal abelian subgroup of order p ³ are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.     

Maximal subgroups of the orthogonal group

Let V be an n-dimensional regular quadratic space over a field K of characteristic not 2. Assume n 4. Let W be a regular hyperplane and v a nonzero vector orthogonal to W. Suppose every regular hyperplane in W is universal. If is an isometry of V not leaving W invariant, then , together with the isometries of W, generate the orthogonal group of V, with one exception.The work of the author was partially supported by NSERC Grant A-7862.