scholarly journals On Groups in Which Many Automorphisms Are Cyclic

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 262
Author(s):  
Mattia Brescia ◽  
Alessio Russo
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Factor Group

Let G be a group. An automorphism α of G is said to be a cyclic automorphism if the subgroup ⟨x,xα⟩ is cyclic for every element x of G. In [F. de Giovanni, M.L. Newell, A. Russo: On a class of normal endomorphisms of groups, J. Algebra and its Applications 13, (2014), 6pp] the authors proved that every cyclic automorphism is central, namely, that every cyclic automorphism acts trivially on the factor group G/Z(G). In this paper, the class FW of groups in which every element induces by conjugation a cyclic automorphism on a (normal) subgroup of finite index will be investigated.

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 Characteristic Morphisms: In Memoriam Thomas Macfarland Cherry

1969 ◽  
Vol 9 (3-4) ◽  
pp. 478-488 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
In Memoriam

This note is concerned with a translation of some concepts and results about characteristic subgroups of a group into the language of categories. As an example, consider strictly characteristic and hypercharacteristic subgroups of a group: the subgroup H of the group G is called strictly characteristic in G if it admits all ependomorphisms of G; that is all homomorphic mappings of G onto G; and H is called hypercharacteristic2 in G if it is the least normal subgroup with factor group isomorphic to G/H, that is if H is contained in every normal subgroup K of G with G/K ≅ G/H.

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 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 The SQ-universality of some finitely presented groups

1973 ◽  
Vol 16 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Peter M. Neumann
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Countable Group ◽  
Factor Group ◽  
Finitely Presented Groups ◽  
Rank 2 ◽  
Finitely Presented

Following a suggestion of G. Higman we say that the group G is SQ-universal if every countable group is embeddable in some factor group of G. It is a well-known theorem of G. Higman, B. H. Neumann and Hanna Neumann that the free group of rank 2 is sq-universal in this sense. Several different proofs are now available (see, for example, [1] or [9]). It is my intention to prove the LEmma. If H is a subgroup of finite index in a group G, then G is SQ-universal if and only if H is SQ-universal.

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).

scholarly journals On some classes of sublattices of the subgroup lattice

2019 ◽  
pp. 35-47 ◽  
Author(s):  
Alexander N. Skiba
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
Subgroup Lattice ◽  
Normal Closure ◽  
Normal Subgroups ◽  
Normal Section ◽  
Normal Core

In this paper G always denotes a group. If K and H are subgroups of G, where K is a normal subgroup of H, then the factor group of H by K is called a section of G. Such a section is called normal, if K and H are normal subgroups of G, and trivial, if K and H are equal. We call any set S of normal sections of G a stratification of G, if S contains every trivial normal section of G, and we say that a stratification S of G is G-closed, if S contains every such a normal section of G, which is G-isomorphic to some normal section of G belonging S. Now let S be any G-closed stratification of G, and let L be the set of all subgroups A of G such that the factor group of V by W, where V is the normal closure of A in G and W is the normal core of A in G, belongs to S. In this paper we describe the conditions on S under which the set L is a sublattice of the lattice of all subgroups of G and we also discuss some applications of this sublattice in the theory of generalized finite T-groups.

scholarly journals Classical groups over division rings of characteristic two

1972 ◽  
Vol 7 (2) ◽  
pp. 191-226 ◽  
Author(s):  
William M. Pender ◽  
G.E. Wall
Keyword(s):  
Normal Subgroup ◽  
Quadratic Form ◽  
Division Ring ◽  
Quadratic Forms ◽  
Factor Group ◽  
Classical Groups ◽  
Isometry Groups ◽  
Division Rings ◽  
Characteristic Two

The notion of quadratic form over a field of characteristic two is extended to an arbitrary division ring of characteristic two with an involution of the first kind. The resulting isometry groups are shown to have a simple normal subgroup and the structure of the factor group is calculated. It is indicated how one may define and analyse all the classical groups in a unified manner by means of quadratic forms.

scholarly journals On supplements in finite groups

1963 ◽  
Vol 3 (1) ◽  
pp. 63-67
Author(s):  
R. Kochendörffer
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Factor Group ◽  
Extension Theory ◽  
A Finite Group

Let G be a finite group. If N denotes a normal subgroup of G, a subgroup S of G is called a supplement of N if we have G = SN. For every normal subgroup of G there is always the trivial supplement S = G. The existence of a non-trivial supplement is important for the extension theory, i.e., for the description of G by means of N and the factor group G/N. Generally, a supplement S is the more useful the smaller the intersection S ∩ N. If we have even S ∩ N = 1, then S is called a complement for N in G. In this case G is a splitting extension of N by S.

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 $Y(g)$.

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