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1.
We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups. 相似文献
2.
Gareth Wilkes 《Journal of Pure and Applied Algebra》2019,223(4):1617-1688
In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer–Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the fundamental group of a compact aspherical 3-manifold is a profinite Poincaré duality group relative to the profinite completions of the fundamental groups of its boundary components. 相似文献
3.
4.
S. G. Melesheva 《Algebra and Logic》2010,49(5):444-455
The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate
group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved
that if the set π(G) of prime divisors of the profinite period of a group G is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups,
the finiteness of a set π(G) gives rise to equational Noetherianness. The concept of a standard linear pro-p-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent
pro-p-groups and free metabelian pro-p-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-p-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined.
For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language
of universal theories and the notion of discriminability. 相似文献
5.
An automorphism of a (profinite) group is called normal if each (closed) normal subgroup is left invariant by it. An automorphism
of an abstract group is p-normal if each normal subgroup of p-power, where p is prime, is left invariant. Obviously, the inner
automorphism of a group will be normal and p-normal. For some groups, the converse was stated to be likewise true. N. Romanovskii
and V. Boluts, for instance, established that for free solvable pro-p-groups of derived length 2, there exist normal automorphisms
that are not inner. Let N2 be the variety of nilpotent groups of class 2 and A the variety of Abelian groups. We prove the following results: (1) If
p is a prime number distinct from 2, then the normal automorphism of a free pro-p-group of rank ≥2 in N2A is inner (Theorem 1); (2) If p is a prime number distinct from 2, then the p-normal automorphism of an abstract free N2A-group of rank ≥2 is inner (Theorem 2).
Supported by RFFR grant No. 93-01-01508.
Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 249–267, May–June, 1996. 相似文献
6.
We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators. 相似文献
7.
P. A. Zalesskii 《Monatshefte für Mathematik》2002,135(2):167-171
A profinite group is said to be just infinite if each of its proper quotients is finite. We address the question which profinite
groups admit just infinite quotients. It is proved that any profinite group whose order (as a supernatural number) is divisible
only by finitely many primes admits just infinite quotients. It is shown that if a profinite group G possesses the property in question then so does every open subgroup and every finite extension of G.
Received 20 July 2001 相似文献
8.
Hiroyuki Nakaoka 《代数通讯》2013,41(9):3095-3151
The Tambara functor was defined by Tambara in the name of TNR-functor, to treat certain ring-valued Mackey functors on a finite group. Recently Brun revealed the importance of Tambara functors in the Witt–Burnside construction. In this article, we define the Tambara functor on the Mackey system of Bley and Boltje. Yoshida's generalized Burnside ring functor is the first example. Consequently, we can consider a Tambara functor on any profinite group. In relation with the Witt–Burnside construction, we can give a Tambara-functor structure on Elliott's functor V M , which generalizes the completed Burnside ring functor of Dress and Siebeneicher. 相似文献
9.
Claus Scheiderer 《manuscripta mathematica》1996,91(1):247-281
LetG be a profinite group which has an open subgroupH such that the cohomologicalp-dimensiond≔cdp(H) is finite (p is a fixed prime). The main result of this paper expresses thep-primary part of high degree cohomology ofG in terms of the elementary abelianp-subgroups ofG: From the latter one constructs a natural profinite simplicial setA
G, on whichG acts by conjugation. ThenH
n(G,M)≅H
G
n
(AG,M) holds forn≧d+r and everyp-primary discreteG-moduleM (r≔p-rank ofG). If one uses profinite Farrell cohomology, which is introduced in this paper, the analogous fact holds in all degrees. These
results are the profinite analogues of theorems by K.S. Brown for discrete groups. 相似文献
10.
We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x, y)) have free absolute Galois groups. 相似文献
11.
Benjamin Steinberg 《Israel Journal of Mathematics》2010,176(1):139-155
We answer a question of Margolis from 1997 by establishing that the maximal subgroup of the minimal ideal of a finitely generated
free profinite monoid is a free profinite group. More generally, if H is variety of finite groups closed under extension and containing ℤ/pℤ for infinitely may primes p, the corresponding result holds for free pro-$
\bar H
$
\bar H
monoids. 相似文献
12.
Let G be a group. An element g ∈ G is called a test element of G if for every endomorphism ? : G → G, ?(g) = g implies ? is an automorphism. We prove that for a finitely generated profinite group G, g ∈ G is a test element of G if and only if it is not contained in a proper retract of G. Using this result we prove that an endomorphism of a free pro-p group of finite rank which preserves an automorphic orbit of a nontrivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-p groups and Demushkin groups. By relating test elements in finitely generated residually finite-p Turner groups to test elements in their pro-p completions, we provide new examples of test elements in free discrete groups and surface groups. Moreover, we prove that the set of test elements of a free discrete group of finite rank is dense in the profinite topology. 相似文献
13.
Young-Tak Oh 《Mathematische Zeitschrift》2007,257(1):151-191
In this paper, we construct a q-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where q ranges over the set of integers. When q = 1, it coincides with the Witt-Burnside ring introduced by Dress and Siebeneicher (Adv. Math. 70, 87–132 (1988)). To achieve
our goal we first show that there exists a q-deformation of the necklace ring of a profinite group over a commutative ring. As in the classical case, i.e., the case q = 1, q-deformed Witt-Burnside rings and necklace rings always come equipped with inductions and restrictions. We also study their
properties. As a byproduct, we prove a conjecture due to Lenart (J. Algebra. 199, 703-732 (1998)). Finally, we classify up to strict natural isomorphism in case where G is an abelian profinite group.
The author gratefully acknowledges support from the following grants: KOSEF Grant # R01-2003-000-10012-0; KRF Grant # 2006-331-C00011. 相似文献
14.
Benjamin Steinberg 《Semigroup Forum》2010,81(1):217-227
We associate a 2-complex to the following data: a presentation of a semigroup S and a transitive action of S on a set V by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex. A sufficient
condition is given for the 2-complex to be simply connected. As a consequence we obtain simple topological proofs of results
on presentations of Schützenberger groups. We also give a geometric proof that a finitely generated regular semigroup with
finitely many idempotents has polynomial growth if and only if all its maximal subgroups are virtually nilpotent. 相似文献
15.
Norman L. Johnson 《Journal of Geometry》1991,40(1-2):95-104
In this article, it is shown that a finite net of orderq
2 and degreeq + 1 is a derivable net if and only if the net admits a collineation group isomorphic toPSL(4,q)N (whereN is a line of the associated 3-dimensional projective space upon which the abstract group acts). 相似文献
16.
A group action is semifree if it is free away from its fixed‐point set. P. A. Smith showed that when a finite group of order q acts semifreely on a sphere, the fixed set is a mod q homology sphere. Conversely, given a mod q homology sphere as a subset of a sphere, one may try to construct a group action on the sphere fixing the subset. The converse question was first systematically studied by Jones and then by many others. In this note, we give new numerical congruences satisfied by the homology of the fixed sets and give a definitive solution to the problem for characteristic fixed‐point sets. © 1999 John Wiley & Sons, Inc. 相似文献
17.
We show that, if the subgroup growth of a finitely generated (abstract or profinite) groupG is super-exponential, then every finite group occurs as a quotient of a finite index subgroup ofG. The proof involves techniques from finite permutation groups, and depends on the Classification of Finite Simple Groups.The first author was partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T7441. The second author was partially supported by the Israeli National Science Foundation. 相似文献
18.
Andrea Lucchini 《Israel Journal of Mathematics》2011,181(1):53-64
It has been conjectured by Mann that the infinite sum Σ
H
μ(H,G)/|G:H|
s
, where H ranges over all open subgroups of a finitely generated profinite group G, converges absolutely in some half right plane if G is positively finitely generated. We prove that the conjecture is true if the nonabelian crowns of G have bounded rank. In particular Mann’s conjecture holds if G has polynomial subgroup growth or is an adelic profinite group. 相似文献
19.
Damien Gaboriau 《Inventiones Mathematicae》2000,139(1):41-98
We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect to finite index subgroups.
Namely, the quantities 1 minus the cost are related by multiplying by the index. The cost of every infinite amenable group
equals 1. We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions
over amenable groups and for direct product situations. For instance, the cost of the free group on n generators equals n. We prove that each possible finite value of the cost is achieved by a finitely generated group. It is dynamical because
it relies on measure preserving free actions on probability Borel spaces. In most cases, groups have fixed price, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us
to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks. At the end of
the paper, we give a mercuriale, i.e. a list of costs of different groups. The cost is in fact an invariant of ergodic measure-preserving equivalence relations
and is defined using graphings. A treeing is a measurable way to provide every equivalence class (=orbit) with the structure of a simplicial tree, this an example
of graphing. Not every relation admits a treeing: we prove that every free action of a cost 1 non-amenable group is not treeable,
but we prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced
by an action of any finitely generated group. The cost of a relation which can be decomposed as a direct product is shown
to be 1. We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting
relation from the costs of the building blocks. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra.
Oblatum 27-I-1999 & 4-IV-1999 / Published online: 22 September 1999 相似文献
20.
Wolfgang Herfort 《manuscripta mathematica》1982,37(1):11-17
We intend to generalize a crucial lemma of [4] to prove a somewhat surprising arithmetic property of profinite groups; namely, that a profinite group G has nontrivial p-Sylow-subgroups for only a finite number of primes if and only if this is true for its procyclic subgroups. This will yield as a corollary that every profinite torsion group has finite exponent if and only if this is true for its Sylow-sub-groups, a result also contained in [4]. 相似文献