Profinite Groups Research Articles

P-elements in profinite groups

p-elements in profinite groups

Minimal compact group actions on C∗-algebras with simple fixed point algebras

The notion of quasi-product actions of a compact group on a C[Formula: see text]-algebra was introduced by Bratteli et al. in their attempt to seek an equivariant analogue of Glimm’s characterization of non-type I C[Formula: see text]-algebras. We show that a faithful minimal action of a second countable compact group on a separable C[Formula: see text]-algebra is quasi-product whenever its fixed point algebra is simple. This was previously known only for compact abelian groups and for profinite groups. Our proof relies on a subfactor technique applied to finite index inclusions of simple C[Formula: see text]-algebras in the purely infinite case, and also uses ergodic actions of compact groups in the general case. As an application, we show that if moreover the fixed point algebra is a Kirchberg algebra, such an action is always isometrically shift-absorbing, and hence is classifiable by the equivariant KK-theory due to a recent result of Gabe-Szabó.

ADJUNCTION OF ROOTS TO PROFINITE GROUPS

In this paper, when $G'$ is a group obtained by adjoining a $n$th-root of $g$ to a given group $G$, where $n$ is a nonzero natural number and $g$ is an element of $G$ of infinite order, we compute the profinite completion $\widehat{G'}$ of $G'$. Also, given $G$ a profinite group in which any subgroup of finite index is open, $n$ a nonzero natural number, $g$ an element of $G$, and $x$ an element not belonging to $G$, we point out necessary and sufficient condition under which the group obtained by adjoining roots to the profinite group $G$ remains again profinite. Our proofs make use of theoretico-combinatorial methods.

On the lattice of closed subgroups of a profinite group

The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behavior of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained.

Integrals of groups. II

AbstractAn integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper continues the investigation on integrals of groups started in the work [1]. We study: A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2). The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups (Theorem 4.1). Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups (Section 5). The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class (Sections 6 and 7). Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups (Section 8.1). Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral (Section 8.2). We end the paper with a number of open problems.

Weil zeta functions of group representations over finite fields

In this article we define and study a zeta function ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document}—similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζG(k)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G(k)^{-1}$$\\end{document} at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document}. We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro-C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {C}}$$\\end{document} groups, where C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {C}}$$\\end{document} is a class of finite groups with prescribed composition factors. We prove that every real number a≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a \\ge 1$$\\end{document} is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} are rational functions in p-s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^{-s}$$\\end{document} if G is virtually abelian. For finite groups G we calculate ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} using the rational representation theory of G.

On profinite groups admitting a word with only few values

On profinite groups admitting a word with only few values

The Haar measure of a profinite n-ary group

We prove that every profinite [Formula: see text]-ary group [Formula: see text] has a unique Haar measure [Formula: see text] and further for every measurable subset [Formula: see text], we have [Formula: see text] where [Formula: see text] and [Formula: see text] are the normalized Haar measures of the profinite groups [Formula: see text] and the Post cover [Formula: see text], respectively.

Solubilizers in profinite groups

The solubilizer of an element x of a profinite group G is the set of the elements y of G such that the subgroup of G generated by x and y is prosoluble. We propose the following conjecture: the solubilizer of x in G has positive Haar measure if and only if x centralizes ‘almost all’ the non-abelian chief factors of G. We reduce the proof of this conjecture to another conjecture concerning finite almost simple groups: there exists a positive c such that, for every finite simple group S and every (a,b)∈(Aut(S)∖{1})×Aut(S), the number of s is S such that 〈a,bs〉 is insoluble is at least c|S|. Work in progress by Fulman, Garzoni and Guralnick is leading to prove the conjecture when S is a simple group of Lie type. In this paper we prove the conjecture for alternating groups.

Large norms in group theory

In 1935, the introduction of the norm of a group by Reinhold Baer is a turning point in group theory. In fact, Baer proved that there is a very strong relationship between the structure of the norm and that of the whole group (see [1], [2], [3], [4], [5]). Since then, the norm has been playing a very significant roles in many aspects of group theory and its applications: it has been used in [43] to describe the connection between Hopf–Galois structures and skew braces; it has been used in [23] to describe some special types of profinite groups; and it has been fundamental in the theory of subgroup lattices of groups (see [40]).In this paper, we weaken the original definition of norm by taking into account only those subgroups that are “large” in some sense. Depending on the chosen concept of largeness, the resulting norm can have an impact on the structure of the whole group that is even greater than that of Baer's norm. This is exactly what happens with the non-polycyclic norm, and in fact, Theorem 4.17 gives a precise description of generalized soluble groups in which the non-polycyclic norm is non-Dedekind (and can be considered as the main result of the paper). Other times, the resulting norms have their own peculiar behaviour; this is the case if “large” means “infinite”, “having infinite rank”, “being non-Černikov”, or “having cardinality m” for some given uncountable cardinal number m.

A homotopy orbit spectrum for profinite groups

A homotopy orbit spectrum for profinite groups

Galois actions of finitely generated groups rarely have model companions

AbstractWe show that if is a finitely generated group such that its profinite completion is “far from being projective” (i.e., the kernel of the universal Frattini cover of is not a small profinite group), then the class of existentially closed ‐actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is “far from being projective,” the main result of this paper corrects an error in our paper, Beyarslan and Kowalski (Proc. London Math. Soc., (2) 118 (2019), 221–256), by showing the negation of Theorem 3.26 in that paper.

Mixed moments and the joint distribution of random groups

We study the joint distribution of random abelian and non-abelian groups. In the abelian case, we prove several universality results for the joint distribution of the multiple cokernels for random p-adic matrices. In the non-abelian case, we compute the joint distribution of random groups given by the quotients of the free profinite group by random relations. In both cases, we generalize the known results on the distribution of the cokernels of random p-adic matrices and random groups. Our proofs are based on the observation that mixed moments determine the joint distribution of random groups, which extends the works of Wood for abelian groups and Sawin for non-abelian groups.

Relatively projective profinite groups

Relatively projective profinite groups

Random generation of associative algebras

AbstractThere has been considerable interest in recent decades in questions of random generation of finite and profinite groups and finite simple groups in particular. In this paper, we study similar notions for finite and profinite associative algebras. Let be a finite field. Let be a finite‐dimensional, associative, unital algebra over . Let be the probability that two elements of chosen (uniformly and independently) at random will generate as a unital ‐algebra. It is known that if is simple, then as . We extend this result to a large class of finite associative algebras. For simple, we find the optimal lower bound for and we estimate the growth rate of in terms of the minimal index of any proper subalgebra of . We also study the random generation of simple algebras by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let be a profinite algebra over . We show that is positively finitely generated if and only if has polynomial maximal subalgebra growth. Related quantitative results are also established.

COMPUTABLE TOPOLOGICAL GROUPS

Abstract We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare our results with similar results in computable topology.

Titchmarsh theorems for Hölder-Lipschitz functions on profinite groups

Titchmarsh theorems for Hölder-Lipschitz functions on profinite groups

Extendible functions and local root numbers: Remarks on a paper of R. P. Langlands

AbstractThis paper refers to Langlands' big set of notes devoted to the question if the (normalized) local Hecke–Tate root number , where E is a finite separable extension of a fixed nonarchimedean local field F, and χ a quasicharacter of , can be appropriately extended to a local ε‐factor for all virtual representations ρ of the corresponding Weil group . Whereas Deligne has given a relatively short proof by using the global Artin–Weil L‐functions, the proof of Langlands is purely local and splits into two parts: the algebraic part to find a minimal set of relations for the functions Δ, such that the existence (and unicity) of will follow from these relations; and the more extensive arithmetic part to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes, which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role.

Profinite groups with abelian Sylow subgroups

We extend the definition of a finite -group to profinite groups and give a description of profinite -groups as a triple semidirect product of two prosoluble groups with a semisimple group, extending an old result of A. M. Broshi to the profinite case. We also prove that a profinite -group with finitely generated non-trivial Fitting subgroup is metabelian-by-(finite exponent). If, in addition, G is finitely generated then it is virtually metabelian polycyclic. Communicated by Mandi Schaeffer Fry

SELF-DIVISIBLE ULTRAFILTERS AND CONGRUENCES IN

AbstractWe introduceself-divisibleultrafilters, which we prove to be precisely those$w$such that the weak congruence relation$\equiv _w$introduced by Šobot is an equivalence relation on$\beta {\mathbb Z}$. We provide several examples and additional characterisations; notably we show that$w$is self-divisible if and only if$\equiv _w$coincides with the strong congruence relation$\mathrel {\equiv ^{\mathrm {s}}_{w}}$, if and only if the quotient$(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$is a profinite group. We also construct an ultrafilter$w$such that$\equiv _w$fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion$\hat {{\mathbb Z}}$of the integers.