共查询到20条相似文献,搜索用时 15 毫秒
1.
Simon Thomas 《Israel Journal of Mathematics》1998,103(1):93-109
We prove that the automorphism tower of every infinite centreless groupG of cardinality κ terminates in less than (2κ)+ steps. We also show that it is consistent withZFC that the automorphism tower of every infinite centreless groupG of regular cardinality κ actually terminates in less than 2κ steps.
Research partially supported by NSF Grants. 相似文献
2.
For a wide class of saturated weakly branch groups, including the (first) Grigorchuk group and the Gupta-Sidki group, we prove
that the Reidemeister number of any automorphism is infinite.
相似文献
3.
Enrico Jabara 《Rendiconti del Circolo Matematico di Palermo》2007,56(3):343-348
LetG be a group and ϕ an automorphism ofG. Two elementsx, y ∈ G are called ϕ-conjugate if there existsg ∈ G such thatx=g
−1
yg
θ. It is easily verified that the ϕ-conjugation is an equivalence relation; the numberR(ϕ) of ϕ-classes ofG is called the Reidemeister number of the automorphism ϕ.
In this paper we prove that if a polycyclic groupsG admits an automorphism ϕ of ordern such thatR(ϕ)<∞, thenG contains a subgroup of finite index with derived length at most 2
n−1
.
相似文献
4.
A. Abdollahi 《代数通讯》2017,45(8):3636-3642
A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper we give some necessary conditions for a possible counterexample G to this conjecture, in the case when G is a 2-generator finite p-group. Then we show that every 2-generator finite p-group with abelian Frattini subgroup has a noninner automorphism of order p. 相似文献
5.
Procopis Psaltis 《Israel Journal of Mathematics》2008,163(1):345-367
A group G is called unsplittable if Hom(G, ℤ) = 0 and this group is not a non-trivial amalgam. Let X be a tree with a countable number of edges incident at each vertex and G be its automorphism group. In this paper we prove that the vertex stabilizers are unsplittable groups.
Bass and Lubotzky proved (see [3]) that for certain locally finite trees X, the automorphism group determines the tree X (that is, knowing the automorphism group we can “construct” the tree X). We generalize this Theorem of Bass and Lubotzky, using the above result. In particular we show that the Theorem holds even
for trees which are not locally finite.
Moreover, we prove that the permutation group of an infinite countable set is unsplittable and the infinite (or finite) cartesian
product of unsplittable groups is an unsplittable group as well.
This research was supported by the European Social Fund and National resources-EPEAEK II grant Pythagoras 70/3/7298. 相似文献
6.
A connected graph G is said to be z-homogeneous if any isomorphism between finite connected induced subgraphs of G extends to an automorphism of G. Finite z-homogeneous graphs were classified in [17]. We show that z-homogeneity is equivalent to finite-transitivity on the class of infinite locally finite graphs. Moreover, we classify the
graphs satisfying these properties. Our study of bipartite z-homogeneous graphs leads to a new characterization for hypercubes. 相似文献
7.
F. K. Indukaev 《Moscow University Mathematics Bulletin》2007,62(6):219-227
For each positive integer N, an automorphism with the Reidemeister number 2N of the discrete Heisenberg group is constructed; an example of determination of points in the unitary dual object being fixed with respect to the mapping induced by the group automorphism is given. For wreath products of finitely generated Abelian groups and the group of integers, it is proved that if the Reidemeister number of an arbitrary automorphism is finite, then it is equal to the number of fixed points of the induced mapping on a finite-dimensional part of the unitary dual object. 相似文献
8.
Florian Lehner 《Combinatorica》2016,36(3):333-347
A graph G is said to be 2-distinguishable if there is a 2-labeling of its vertices which is not preserved by any nontrivial automorphism of G. We show that every locally finte graph with infinite nite motion and growth at most \(\mathcal{O}\left( {2^{(1 - \varepsilon )\tfrac{{\sqrt n }}{2}} } \right)\) is 2-distinguishable. Infinite motion means that every automorphism moves infinitely many vertices and growth refers to the cardinality of balls of radius n. 相似文献
9.
Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is developed further here.¶ Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no infinite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group.¶ We show that G is amenable if for all $ \alpha < 1 $ \alpha < 1 , there is a G-invariant site percolation process w \omega on X with $ {\bf P} [x \in \omega] > \alpha $ {\bf P} [x \in \omega] > \alpha for all vertices x and with no infinite components. When G is not amenable, a threshold $ \alpha < 1 $ \alpha < 1 appears. An inequality for the threshold in terms of the isoperimetric constant is obtained, extending an inequality of Häggström for regular trees.¶ If G acts transitively on X, we show that G is unimodular if the expected degree is at least 2 in any G-invariant bond percolation on X with all components infinite.¶ The investigation of dependent percolation also yields some results on automorphism groups of graphs that do not involve percolation. 相似文献
10.
In this note we show that if Gis a finitely generated soluble group, then every infinite subset of Gcontains two elements generating a nilpotent group of class at most kif and only if Gis finite by a group in which every two generator subgroup is nilpotent of class at most k. 相似文献
11.
Enrico Jabara 《Rendiconti del Circolo Matematico di Palermo》2005,54(3):367-380
An automorphismϕ of a groupG is said to be uniform il for everyg ∈G there exists anh ∈G such thatG=h
−1
h
ρ
. It is a well-known fact that ifG is finite, an automorphism ofG is uniform if and only if it is fixed-point-free. In [7] Zappa proved that if a polycyclic groupG admits an uniform automorphism of prime orderp thenG is a finite (nilpotent)p′-group.
In this paper we continue Zappa’s work considering uniform automorphism of orderpg (p andq distinct prime numbers). In particular we prove that there exists a constantμ (depending only onp andq) such that every torsion-free polycyclic groupG admitting an uniform automorphism of orderpq is nilpotent of class at mostμ. As a consequence we prove that if a polycyclic groupG admits an uniform automorphism of orderpq thenZ
μ
(G) has finite index inG.
Al professore Guido Zappa per il suo 900 compleanno 相似文献
12.
13.
N. Polat 《Periodica Mathematica Hungarica》1993,27(2):125-136
It is shown that ifG is a graph which is contractible or dismantlable or finitely ball-Helly, and without infinite paths; or which is bounded, finitely ball-Helly and without infinite simplices then: (i) any contraction ofG stabilizes a finite simplex; and (ii)G contains a finite simplex which is invariant under any automorphism. 相似文献
14.
A group G is said to have the Bergman property (the propertyof uniformity of finite width) if given any generating X withX = X1 of G, we have that G = Xk for some natural k,that is, every element of G is a product of at most k elementsof X. We prove that the automorphism group Aut(N) of any infinitelygenerated free nilpotent group N has the Bergman property. Also,we obtain a partial answer to a question posed by Bergman byestablishing that the automorphism group of a free group ofcountably infinite rank is a group of uniformly finite width. 相似文献
15.
Ngo Dac Tuan 《Journal of Combinatorial Theory, Series A》2002,100(2):403
We give a new recursive construction of simple non-trivial designs. Using this construction, we show that given a natural number t and a finite group G, a simple non-trivial t-design admitting an automorphism group isomorphic to G exists. Further, we apply our construction to get a recursive construction of large sets. 相似文献
16.
We give a practical criterion to determine whether a given pair of morphisms between almost-crystallographic groups has a
finite Reidemeister coincidence number. As an application, we determine all two- and three-dimensional almost-crystallographic
groups that have the R
∞ property. We also show that for a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, the Nielsen
coincidence number equals the Reidemeister coincidence number when the latter is finite. 相似文献
17.
A graph G is called strongly p-reconstructible if it is (up to isomorphism) uniquely determined by the collection of its pairwise nonisomorphic subgraphs G – v where v is a pendent vertex of G. Using previous results of the second author concerning the structure of infinite rayless graphs it is shown that every rayless forest with an infinite edge-set is strongly p-reconstructible. This result is applied to the classical reconstruction problem to find that every infinite rayless forest G with a finite number of components is reconstructible; i.e., G is (up to isomorphism) uniquely determined by its collection of vertex-deleted subgraphs. Furthermore, an example is given which shows that nonreconstructible rayless forests with a countable number of components exist. 相似文献
18.
Wilfried Imrich Simon M. Smith 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2017,87(2):289-297
This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph G, with \(\aleph _0 \le |{\text {Aut}}(G)| < 2^{\aleph _0}\) and subdegree-finite automorphism group, has a finite set F of vertices that is setwise stabilized only by the identity automorphism. A bound on the size of such sets, which are called distinguishing, is also provided. To put this theorem of Halin and its generalization into perspective, we also discuss several related non-elementary, independent results and their methods of proof. 相似文献
19.
E. I. Bunina 《Journal of Mathematical Sciences》2008,155(6):795-814
In the paper, we prove that every automorphism of any adjoint Chevalley group of type B
2 or G
2 is standard, i.e., it is a composition of an “inner” automorphism, a ring automorphism, and a central automorphism.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 3–29, 2007. 相似文献
20.
In (Electron. J. Combin. 10 (2003); http://www.combinatorics.org/volume-10/Abstracts/v1oi1r28.html), the first author (Yuliya Gryshko) asked three questions. Is it true that every infinite group admitting a 2-coloring without infinite monochromatic symmetric subsets is either almost cyclic (i.e., have a finite index subgroup which is cyclic infinite) or countable locally finite? Does every infinite group G include a monochromatic symmetric subset of any cardinal <|G| for any finite coloring? Does every uncountable group G such that |B(G)|< |G| where B(G)={xG:x2=1}, admit a 2-coloring without monochromatic symmetric subsets of cardinality |G|? We answer the first question positively. Assuming the generalized continuum hypothesis (GCH), we give a positive answer to the second question in the abelian case. Finally, we build a counter-example for the third question and we give a necessary and sufficient condition for an infinite group G to admit 2-coloring without monochromatic symmetric subsets of cardinality |G|. This generalizes some results of Protasov on infinite abelian groups (Mat. Zametki 59 (1996) 468–471; Dopovidi NAN Ukrain 1 (1999) 54–57). 相似文献