The Experts below are selected from a list of 291 Experts worldwide ranked by ideXlab platform
Pavel Shumyatsky - One of the best experts on this subject based on the ideXlab platform.
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Profinite Groups in which many elements have prime power order
arXiv: Group Theory, 2020Co-Authors: Pavel ShumyatskyAbstract:The structure of finite and locally finite Groups in which every element has prime power order (CP-Groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of finite CP-Groups yields a detailed description of Profinite Groups with that property. Then we deal with two generalizations of Profinite CP-Groups. Theorem 1.2. A Profinite Group G is virtually pro-p for some prime p if and only if for each nontrivial x in G there is a prime p (depending on x) such that the centralizer of x is virtually pro-p. Theorem 1.3. Let G be a Profinite Group in which each element has either finite or prime power (possibly infinite) order. Then G is either torsion or virtually pro-p for some prime p.
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Profinite Groups in which centralizers are virtually procyclic
arXiv: Group Theory, 2019Co-Authors: Pavel Shumyatsky, Pavel ZalesskiiAbstract:The article deals with Profinite Groups in which centralizers are virtually procyclic. Suppose that G is a Profinite Group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infinite order is virtually procyclic. We show that G is either virtually pro-p for some prime p or virtually torsion-free procyclic. The same conclusion holds for Profinite Groups in which the centralizer of every nontrivial element is virtually procyclic; moreover, if G is not pro-p, then G has finite rank.
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Strong conciseness in Profinite Groups
arXiv: Group Theory, 2019Co-Authors: Eloisa Detomi, Benjamin Klopsch, Pavel ShumyatskyAbstract:A Group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of Profinite Groups if, for every Group $G$ in $\mathcal{C}$ such that $w$ takes less than $2^{\aleph_0}$ values in $G$, the verbal subGroup $w(G)$ is finite. Detomi, Morigi and Shumyatsky established that multilinear commutator words -- and the particular words $x^2$ and $[x^2,y]$ -- have the property that the corresponding verbal subGroup is finite in a Profinite Group $G$ whenever the word takes at most countably many values in $G$. They conjectured that, in fact, this should be true for every word. In particular, their conjecture included as open cases power words and Engel words. In the present paper, we take a new approach via parametrised words that leads to stronger results. First we prove that multilinear commutator words are strongly concise in the class of all Profinite Groups. Then we establish that every Group word is strongly concise in the class of nilpotent Profinite Groups. From this we deduce, for instance, that, if $w$ is one of the Group words $x^2$, $x^3$, $x^6$, $[x^3,y]$ or $[x,y,y]$, then $w$ is strongly concise in the class of all Profinite Groups. Indeed, the same conclusion can be reached for all words of the infinite families $[x^m,z_1,\ldots,z_r]$ and $[x,y,y,z_1,\ldots,z_r]$, where $m \in \{2,3\}$ and $r \ge 1$.
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Profinite Groups with restricted centralizers of commutators
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2019Co-Authors: Eloisa Detomi, Marta Morigi, Pavel ShumyatskyAbstract:AbstractA Group G has restricted centralizers if for each g in G the centralizer $C_G(g)$ either is finite or has finite index in G. A theorem of Shalev states that a Profinite Group with restricted centralizers is abelian-by-finite. In the present paper we handle Profinite Groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a Profinite Group with restricted centralizers of w-values, then the verbal subGroup w(G) is abelian-by-finite.
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On Profinite Groups with commutators covered by countably many cosets
arXiv: Group Theory, 2017Co-Authors: Eloisa Detomi, Marta Morigi, Pavel ShumyatskyAbstract:Let w be a Group-word. Suppose that the set of all w-values in a Profinite Group G is contained in a union of countably many cosets of subGroups. We are concerned with the question to what extent the structure of the verbal subGroup w(G) depends on the properties of the subGroups. We prove the following theorem. Let C be a class of Groups closed under taking subGroups, quotients, and such that in any Group the product of finitely many normal C-subGroups is again a C-subGroup. If w is a multilinear commutator and G is a Profinite Group such that the set of all w-values is contained in a union of countably many cosets g_iG_i where each G_i is in C, then the verbal subGroup w(G) is virtually-C. This strengthens several known results.
Ido Efrat - One of the best experts on this subject based on the ideXlab platform.
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the lower p central series of a free Profinite Group and the shuffle algebra
Journal of Pure and Applied Algebra, 2020Co-Authors: Ido EfratAbstract:Abstract For a prime number p and a free Profinite Group S on the basis X, let S ( n , p ) , n = 1 , 2 , … , be the lower p-central filtration of S. For p > n , we give a combinatorial description of H 2 ( S / S ( n , p ) , Z / p ) in terms of the Shuffle algebra on X.
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the p zassenhaus filtration of a free Profinite Group and shuffle relations
arXiv: Number Theory, 2020Co-Authors: Ido EfratAbstract:For a prime number $p$ and a free Profinite Group $S$ on the basis $X$, let $S_{(n,p)}$, $n=1,2,\ldots,$ be the $p$-Zassenhaus filtration of $S$. For $p>n$, we give a word-combinatorial description of the cohomology Group $H^2(S/S_{(n,p)},\mathbb{Z}/p)$ in terms of the shuffle algebra on $X$. We give a natural linear basis for this cohomology Group, which is constructed by means of unitriangular representations arising from Lyndon words.
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The Zassenhaus filtration, Massey products, and representations of Profinite Groups
Advances in Mathematics, 2014Co-Authors: Ido EfratAbstract:Abstract We consider the p-Zassenhaus filtration ( G n ) of a Profinite Group G. Suppose that G = S / N for a free Profinite Group S and a normal subGroup N of S contained in S n . Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p-cohomological dimension ≤ 1), we prove that G n + 1 is the intersection of all kernels of upper-triangular unipotent ( n + 1 ) -dimensional representations of G over F p . This extends earlier results by Minac, Spira, and the author on the structure of absolute Galois Groups of fields.
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The Zassenhaus filtration, Massey Products, and Representations of Profinite Groups
arXiv: Number Theory, 2013Co-Authors: Ido EfratAbstract:We consider the p-Zassenhaus filtration (G_n) of a Profinite Group G. Suppose that G=S/N for a free Profinite Group S and a normal subGroup N of S contained in S_n. Under a cohomological assumption on the n-fold Massey products (which holds e.g., if the p-cohomological dimension of G is at most 1), we prove that G_{n+1} is the intersection of all kernels of upper-triangular unipotent (n+1)-dimensional representations of G over \mathbb F_p. This extends earlier results by Minac, Spira, and the author on the structure of absolute Galois Groups of fields.
Colin D. Reid - One of the best experts on this subject based on the ideXlab platform.
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On endomorphisms of Profinite Groups
Groups Geometry and Dynamics, 2014Co-Authors: Colin D. ReidAbstract:We obtain some general restrictions on the continuous endomorphisms of a Profinite Group G under the assumption that G has only finitely many open subGroups of each index (an assumption which automatically holds, for instance, if G is finitely generated). In particular, given such a
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the Profinite completion of a Group localised at a subGroup
arXiv: Group Theory, 2014Co-Authors: Colin D. ReidAbstract:Let G be a Group and let K be a commensurated subGroup of G. Then there is a totally disconnected, locally compact (t.d.l.c.) Group ˆ GK that contains the Profinite completion of K as an open compact subGroup and also contains G (modulo the finite residual of K) as a dense subGroup. Moreover, given an arbitrary Group G, then every t.d.l.c. Group containing an image of G as a dense subGroup can be realised as a quotient of ˆ GK for some commensurated subGroup K. The Profinite completion is a natural way of embedding a residually finite Group in a Profinite Group. Some properties of the original Group can usefully be analysed through its Profinite completion, since Profinite Groups are well-behaved with respect to finite images, and as compact Groups, they are amenable to analytic methods. (See [5] for some examples of Profinite methods in Group theory.) This approach can be generalised to other classes of Group that may not themselves have interesting finite images, but nevertheless have a residually finite subGroup that is commensurated under conjugation, for instance SLn(Z) as a subGroup of SLn(Q). In this setting, we can obtain a relative or localised Profinite completion that is a totally disconnected, locally compact (t.d.l.c.) Group G, such that the image of the commensurated subGroup is dense in an open compact subGroup of G. This approach has been taken, for instance, in [4] and [6].
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the generalised pro fitting subGroup of a Profinite Group
Communications in Algebra, 2013Co-Authors: Colin D. ReidAbstract:The generalised Fitting subGroup of a finite Group is the Group generated by all subnormal subGroups that are either nilpotent or quasisimple. The importance of this subGroup in finite Group theory stems from the fact that it always contains its own centraliser, so that any finite Group is an abelian extension of a Group of automorphisms of its generalised Fitting subGroup. We define a class of Profinite Groups which generalises this phenomenon, and explore some consequences for the structure of Profinite Groups.
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On endomorphisms of Profinite Groups
arXiv: Group Theory, 2011Co-Authors: Colin D. ReidAbstract:We obtain some general restrictions on the continuous endomorphisms of a Profinite Group G under the assumption that G has only finitely many open subGroups of each index (an assumption which automatically holds, for instance, if G is finitely generated). In particular, given such a Group G and a continuous endomorphism phi we obtain a semidirect decomposition of G into a 'contracting' normal subGroup and a complement on which phi induces an automorphism; both the normal subGroup and the complement are closed. If G is isomorphic to a proper open subGroup of itself, we show that G has an infinite abelian normal pro-p subGroup.
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On the structure of just infinite Profinite Groups
Journal of Algebra, 2010Co-Authors: Colin D. ReidAbstract:Abstract 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 (2009) in [1] , who proved the same characterisation in the case of pro- p Groups. We also use this result to establish a number of features of the general structure of Profinite Groups with regard to the just infinite property.
Lior Barysoroker - One of the best experts on this subject based on the ideXlab platform.
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diamond theorem for a finitely generated free Profinite Group
Mathematische Annalen, 2006Co-Authors: Lior BarysorokerAbstract:We extend Haran’s Diamond Theorem to closed subGroups of a finitely generated free Profinite Group. This gives an affirmative answer to Problem 25.4.9 of Fried and Jarden (in Field Arithmetic, 2nd edn, Springer, Berlin Heidelberg, New York, 2005).
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diamond theorem for finitely generated Profinite free Group
arXiv: Number Theory, 2005Co-Authors: Lior BarysorokerAbstract:We extend Haran's Diamond Theorem to closed subGroups of a finitely generated free Profinite Group. This gives an affirmative answer to Problem 25.4.9 in the book Field Arithmetics of Fried and Jarden.
Mark Shusterman - One of the best experts on this subject based on the ideXlab platform.
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an axiomatizable Profinite Group with infinitely many open subGroups of index 2
arXiv: Group Theory, 2016Co-Authors: Or Ben Porath, Mark ShustermanAbstract:We show that a Profinite Group with the same first-order theory as the direct product over all odd primes $p$ of the dihedral Group of order $2p$, is necessarily isomorphic to this direct product.
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Free subGroups of finitely generated free Profinite Groups
Journal of The London Mathematical Society-second Series, 2016Co-Authors: Mark ShustermanAbstract:We give new and improved results on the freeness of subGroups of free Profinite Groups: A subGroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subGroup of a finitely generated nonabelian free Profinite Group is contained in an infinitely generated free Profinite subGroup. These results are combined with the twisted wreath product approach of Haran, an observation on the action of compact Groups, and a rank counting argument to prove a conjecture of Bary-Soroker, Fehm, and Wiese, thus providing a quite general sufficient condition for subGroups to be free Profinite. As a result of our work, we are able to address a conjecture of Jarden on the Hilbertianity of fields generated by torsion points of abelian varieties.
