scholarly journals Averaging formula for Nielsen coincidence numbers

2007 ◽  
Vol 186 ◽  
pp. 69-93 ◽  
Author(s):  
Seung Won Kim ◽  
Jong Bum Lee
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Holonomy Group ◽  
Smooth Manifolds

AbstractIn this paper we study the averaging formula for Nielsen coincidence numbers of pairs of maps (f,g): M→N between closed smooth manifolds of the same dimension. Suppose that G is a normal subgroup of Π = π1(M) with finite index and H is a normal subgroup of Δ = π1(N) with finite index such that Then we investigate the conditions for which the following averaging formula holdswhere is any pair of fixed liftings of (f, g). We prove that the averaging formula holds when M and N are orientable infra-nilmanifolds of the same dimension, and when M = N is a non-orientable infra-nilmanifold with holonomy group ℤ2 and (f, g) admits a pair of liftings on the nil-covering of M.

scholarly journals Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

2021 ◽  
Author(s):  
Jiahao Qiu ◽  
Jianjie Zhao
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Minimal System ◽  
Combinatorial Properties ◽  
Return Times ◽  
Residual Set ◽  
Closed Sets ◽  
Open Set ◽  
Geometric Progressions ◽  
Set Of Points

AbstractIn this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

scholarly journals On groups with all subgroups almost subnormal

2004 ◽  
Vol 77 (2) ◽  
pp. 165-174 ◽  
Author(s):  
Eloisa Detomi
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Finite Index ◽  
Normal Closure ◽  
Fixed Integer ◽  
Torsion Free

AbstractIn this paper we consider groups in which every subgroup has finite index in the nth term of its normal closure series, for a fixed integer n. We prove that such a group is the extension of a finite normal subgroup by a nilpotent group, whose class is bounded in terms of n only, provided it is either periodic or torsion-free.

scholarly journals GIBBS MEASURES FOR SOS MODELS ON A CAYLEY TREE

2006 ◽  
Vol 09 (03) ◽  
pp. 471-488 ◽  
Author(s):  
U. A. ROZIKOV ◽  
Y. M. SUHOV
Keyword(s):  
Ising Model ◽  
Normal Subgroup ◽  
Finite Index ◽  
Cayley Tree ◽  
Nearest Neighbor ◽  
Gibbs Measures ◽  
Natural Generalization ◽  
Mirror Reflection ◽  
Sos Model ◽  
Translation Invariant

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values 0, 1,…, m, m ≥ 2, and zero external field, on a Cayley tree of order k (with k + 1 neighbors). The SOS model can be treated as a natural generalization of the Ising model (obtained for m = 1). We mainly assume that m = 2 (three spin values) and study translation-invariant (TI) and "splitting" (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) Furthermore, we focus on symmetric TISGMs, with respect to a "mirror" reflection of the spins. [For the Ising model (where m = 1), such measures are reduced to the "disordered" phase obtained for free boundary conditions, see Refs. 9, 10.] For m = 2, in the antiferromagnetic (AFM) case, a symmetric TISGM (and even a general TISGM) is unique for all temperatures. In the ferromagnetic (FM) case, for m = 2, the number of symmetric TISGMs and (and the number of general TISGMs) varies with the temperature: this gives an interesting example of phase transition. Here we identify a critical inverse temperature, [Formula: see text] such that [Formula: see text], there exists a unique symmetric TISGM μ* and [Formula: see text] there are exactly three symmetric TISGMs: [Formula: see text] (a "bottom" symmetric TISGM), [Formula: see text] (a "middle" symmetric TISGM) and [Formula: see text] (a "top" symmetric TISGM). For [Formula: see text] we also construct a continuum of distinct, symmertric SGMs which are non-TI. Our second result gives complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree. A complete description of periodic GMs means a characterisation of such measures with respect to any given normal subgroup of finite index in the representation group of the tree. We show that (i) for an FM SOS model, for any normal subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an AFM SOS model, for any normal subgroup of finite index, each periodic SGM is either TI or has period two (i.e. is a chess-board SGM).

A NILPOTENCY-LIKE CONDITION FOR INFINITE GROUPS

2018 ◽  
Vol 105 (1) ◽  
pp. 24-33
Author(s):  
M. DE FALCO ◽  
F. DE GIOVANNI ◽  
C. MUSELLA ◽  
N. TRABELSI
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Finite Index ◽  
Conjugacy Classes ◽  
Nilpotent Subgroup ◽  
Infinite Groups ◽  
Finite Conjugacy

If $k$ is a positive integer, a group $G$ is said to have the $FE_{k}$-property if for each element $g$ of $G$ there exists a normal subgroup of finite index $X(g)$ such that the subgroup $\langle g,x\rangle$ is nilpotent of class at most $k$ for all $x\in X(g)$. Thus, $FE_{1}$-groups are precisely those groups with finite conjugacy classes ($FC$-groups) and the aim of this paper is to extend properties of $FC$-groups to the case of groups with the $FE_{k}$-property for $k>1$. The class of $FE_{k}$-groups contains the relevant subclass $FE_{k}^{\ast }$, consisting of all groups $G$ for which to every element $g$ there corresponds a normal subgroup of finite index $Y(g)$ such that $\langle g,U\rangle$ is nilpotent of class at most $k$, whenever $U$ is a nilpotent subgroup of class at most $k$ of $Y(g)$.

scholarly journals A Note on Regular Coverings of Closed Orientable Surfaces

1961 ◽  
Vol 5 (2) ◽  
pp. 49-66 ◽  
Author(s):  
Jens Mennicke
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Finite Index ◽  
Orientable Surface ◽  
Fundamental Groups ◽  
Closed Orientable Surface ◽  
Regular Covering ◽  
Genus 2 ◽  
Image Position ◽  
Orientable Surfaces

The object of this note is to study the regular coverings of the closed orientable surface of genus 2.Let the closed orientable surfaceFhof genushbe a covering ofF2and letand f be the fundamental groups respectively. Thenis a subgroup of f of indexn = h − 1. A covering is called regular ifis normal in f.Conversely, letbe a normal subgroup of f of finite index. Then there is a uniquely determined regular coveringFhsuch thatis the fundamental group ofFh. The coveringFhis an orientable surface. Since the indexnofin f is supposed to be finite,Fhis closed, and its genus is given byn=h − 1.The fundamental group f can be defined by.

Groups in which each subnormal subgroup is commensurable with some normal subgroup

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ulderico Dardano ◽  
Fausto De Mari
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Soluble Group ◽  
Study Groups ◽  
Subnormal Subgroup

Abstract We study groups in which each subnormal subgroup is commensurable with a normal subgroup. Recall that two subgroups 𝐻 and 𝐾 are termed commensurable if H ∩ K H\cap K has finite index in both 𝐻 and 𝐾. Among other results, we show that if a (sub)soluble group 𝐺 has the above property, then 𝐺 is finite-by-metabelian, i.e., G ′′ G^{\prime\prime} is finite.

Free Groups and Folding

2017 ◽  
Author(s):  
Matt Clay
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Membership Problem ◽  
Research Projects ◽  
Residually Finite ◽  
Simple Operation ◽  
Nontrivial Normal Subgroup ◽  
Subgroup Theorem

This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show the rank of the free group H and whether every finitely generated nontrivial normal subgroup of a free group has finite index. The edge paths and the fundamental group of a graph are discussed, along with subgroups via graphs. The chapter also considers five applications of folding: the Nielsen–Schreier Subgroup theorem, the membership problem, index, normality, and residual finiteness. A group G is residually finite if for every nontrivial element g of G there is a normal subgroup N of finite index in G so that g is not in N. Exercises and research projects are included.

An Induction Theorem for Units of p-Adic Group Rings

1991 ◽  
Vol 34 (1) ◽  
pp. 31-35 ◽  
Author(s):  
A. K. Bhandari ◽  
S. K. Sehgal
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Index ◽  
Group Rings ◽  
Cyclic Subgroups ◽  
The Family ◽  
A Finite Group

AbstractLet G be a finite group and let C be the family of cyclic subgroups of G. We show that the normal subgroup H of U = U(ZpG) generated by U(ZpC), C ∊ C, where Zp is the ring of p-adic integers, is of finite index in U.

scholarly journals COHOMOLOGY AND PROFINITE TOPOLOGIES FOR SOLVABLE GROUPS OF FINITE RANK

2012 ◽  
Vol 86 (2) ◽  
pp. 254-265 ◽  
Author(s):  
KARL LORENSEN
Keyword(s):  
Normal Subgroup ◽  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Graphic Module

AbstractAssume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to Cp∞. We show that if G is nilpotent, then the pro-p completion map $G\to \hat {G}_p$ induces an isomorphism $H^\ast (\hat {G}_p,M)\to H^\ast (G,M)$ for any discrete $\hat {G}_p$-module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map $H^\ast (\hat {N}_p,M)\to H^\ast (N,M)$ is an isomorphism for any discrete $\hat {N}_p$-module M of finite p-power order. Moreover, if G lacks any Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

Some Characterizations of a Normal Subgroup of a Group and Isotopic Classes of Transversals

2016 ◽  
Vol 23 (03) ◽  
pp. 409-422 ◽  
Author(s):  
Vipul Kakkar ◽  
R. P. Shukla
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Dihedral Group ◽  
Elementary Proof ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Right Loop

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this paper, we give some characterizations for the normality of H in G. As a consequence we get a very short and elementary proof of the main theorem of a paper of Lal and Shukla, which avoids the use of the classification of finite simple groups. Further, we study the isotopy between the transversals in some groups and determine the number of isotopy classes of transversals of a subgroup of order 2 in D2p, the dihedral group of order 2p, where p is an odd prime and the isotopism classes are formed with respect to induced right loop structures.

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