A profinite group G is just infinite if every closed normal subgroup of G is of finite index. We prove that an infinite profinite group is just infinite if and only if, for every open subgroup H of G, there are only finitely many open normal subgroups of G not contained in H . This extends a result recently established by Barnea, Gavioli, Jaikin-Zapirain, Monti and Scoppola in [1], who proved t...

We show that the quasiconvex subgroups in doubles of certain negatively curved groups are closed in the profinite topology. This allows us to construct the first known large family of hyperbolic 3-manifolds such that any finitely generated subgroup of the fundamental group of any member of the family is closed in the profinite topology.

The theory of profinite groups is flourishing! This is the first immediate observation from looking at the four books on the subject which have come out in the last two years. The subject, which only two decades ago was somewhat remote, has made its way to mainstream mathematics in several different ways. What is a profinite group? A profinite group G is a topological group which is Hausdorff, ...

in this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid g(n) . also we show that g(n) contains commuting regular subsemigroup and give a necessary and sucient condition for the groupoid g(n) to be commuting regular.

We propose a model-theoretic framework for investigating profinite structures. We prove that in many cases small profinite structures interpret infinite groups. This corresponds to results of Hrushovski and Peterzil on interpreting groups in locally modular stable and o-minimal structures.

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields.

The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially classified up to homeomorphism.

We establish the following model-theoretic characterization: profinite L-structures, the cofiltered limits of finite L-structures, are retracts of ultraproducts of finite L-structures. As a consequence, any elementary class of L-structures axiomatized by L-sentences of the form ∀~x(ψ0(~x) → ψ1(~x)), where ψ0(~x), ψ1(~x) are existencialpositives L-formulas, is closed under the formation of profi...