Gabor P. Nagy
In this paper, we determine the collineation groups generated by the Bol reflections, the core, the automorphism groups and the full direction preserving collineation groups of the loops $B_{4n}$ and $C_{4n}$ given by R.P. Burn. We also prove some lemmas and use new methods in order to simplify the calculations in these groups.
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The Burns-Hale theorem states that a group G is left-orderable if and only if G is locally projectable onto the class of left-orderable groups. Similar results have appeared in the literature in the case of UPP groups and Conradian left-orderable groups, with proofs using varied techniques in each case. This note presents a streamlined approach to showing that if C is the class of either Conradian left-orderable, left-orderable, or UPP groups, then C contains all groups that are locally projectable onto C; and shows that this streamlined approach works for the class of diffuse groups as well. …
The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and S(f)=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays an important role in the theory of twisted conjugacy classes and has very important consequences in Dyn…
Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.
Let $R(ϕ)$ be the number of $ϕ$-conjugacy (or Reidemeister) classes of an endomorphism $ϕ$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(ϕ)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(ϕ)<\infty$. Applying the result to iterations of $ϕ$ we obtain Gauss congruences for Reidemeister numbers. In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_\infty$ pr…
We prove that the restricted wreath product ${\mathbb{Z}_n \mathbin{\mathrm{wr}} \mathbb{Z}^k}$ has the $R_\infty$-property, i. e. every its automorphism $φ$ has infinite Reidemeister number $R(φ)$, in exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$ divisible by 3. In the remaining cases there are automorphisms with finite Reidemeister number, for which we prove the finite-dimensional twisted Burnside--Frobenius theorem (TBFT): $R(φ)$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action ${[ρ]\mapsto…
The purpose of the present paper is to discuss the following conjecture of Fel'shtyn and Hill, which is a generalization of the classical Burnside theorem: Let G be a countable discrete group, f its automorphism, R(f) the number of f-conjugacy classes (Reidemeister number), S(f):=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history and has very serious consequences in Dynamics, …
Chirality and Racemization on Isotopy Classes of Loops: A Groupoid-Based Structural Theory
本文提出基于群体的回旋性理论，通过回旋同伦类的双态动力学模型，揭示回旋性判据与镜像同伦的存在关系。
We classify twisted conjugacy classes in loop groups, restricted to classical groups. The main tool we used is the so-called D_q module, an object which is related to vector bundles over elliptic curves.
Let $G$ be a finite group and $H$ a subgroup of $G$. Each left transversal (with identity) of $H$ in $G$ has a left loop (left quasigroup with identity) structure induced by the binary operation of $G$. We say two left transversals are isomorphic if they are isomorphic with respect to the induced left loop structures. In this paper, we develop a method to calculate the number of isomorphism classes of transversals of $H$ in $G$. Also with the help of this we calculate the number of non-isomorphic left loops of a given order.
The gyrogroup is the closest algebraic structure to the group ever discovered. It has a binary operation $\star$ containing an identity element such that each element has an inverse. Furthermore, for each pair $(a,b)$ of elements of this structure there exists an automorphism $\gyr{a,b}{}$ with this property that left associativity and left loop property are satisfied. Since each gyrogroup is a left Bol loop, some results of Burn imply that all gyrogroups of orders $p, 2p$ and $p^2$ are groups. The aim of this paper is to classify gyrogroups of orders 8, 12, 15, 18, 20, 21, and 28.
On Conjugacy Classes of Derangements in Symmetric and Alternating Groups
本文证明了对称群与交错群中错位排列共轭类的两个猜想，还肯定回答了伯特伦的特定问题，并得出错位排列共轭类生成对称或交错群的相关结论。
In 1911, Burnside asked whether or not there exist groups that have an outer automorphism which preserves conjugacy classes. Two years later he answered his own question by constructing a family of such groups. Using the small group library in MAGMA we determine all of the groups of order n < 512 that possess such an automorphism. Our investigations led to the discovery of four new infinite families of such groups, all of which are 2-groups of coclass 4.
In this short paper, we survey the results on commutative automorphic loops and give a new construction method. Using this method, we present new classes of commutative automorphic loops of exponent 2 with trivial center.
Concise formulae in groups of non-positive curvature
研究非正曲率群及扩展群中一阶公式的简洁性，探讨各类群的公式的简洁性及可定义集合的有限性。
We prove that for any automorphism $ϕ$ of the restricted wreath product $\mathbb{Z}_2 \mathrm{wr} \mathbb{Z}^k$ and $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d}$ the Reidemeister number $R(ϕ)$ is infinite, i.e. these groups have the property $R_\infty$. For $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d+1}$ and $\mathbb{Z}_p \mathrm{wr} \mathbb{Z}^k$, where $p>3$ is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property $R_\infty$. For these groups and $\mathbb{Z}_m \mathrm{wr} \mathbb{Z}$, where $m$ is relatively prime to $6$, we prove the t…
In this article we determine the number of isotopic classes of transversals of a subgroup of order 2 in $ D_{2n}$ ($n$ is a positive odd integer greater than 1), where isotopism classes are formed with respect to the induced right loop structures. We also determine the cyclic index of the Affine group Aff$(1,p^{2})$, where $p$ is an odd prime.
A loop whose inner mappings are automorphisms is an \emph{automorphic loop} (or \emph{A-loop}). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate the classification of commutative A-loops of small orders and also of order $p^3$, where $p$ is a prime.
Finite groups with conjugacy classes of size $2p^n$
该研究推广了Burnside $p^a$定理，证明非阿贝尔有限单群无大小为$2p^n$的共轭类，并推广Kazarin结果，明确有限群中特定大小共轭类生成子群的可解性及非可解时的结构。
W. Goldman and V. Turaev defined a Lie bialgebra structure on the $\mathbb Z$-module generated by free homotopy classes of loops of an oriented surface (i.e. the conjugacy classes of its fundamental group). We develop a generalization of this construction replacing homotopies by thin homotopies, based on the combinatorial approach given by M. Chas. We use it to give a geometric proof of a characterization of simple curves in terms of the Goldman-Turaev bracket, which was conjectured by Chas.
We describe the structure of the algebraic group of automorphisms of all simple finite dimensional Lie superalgebras. Using this and étale cohomology considerations, we list all different isomorphism classes of the corresponding twisted loop superalgebras.
Reidemeister classes in some wreath products by $\mathbb Z^k$
本文在 $G$ 为有限 Abelian 群的受限圈积 $G\wr \mathbb Z^k$ 中找到三大类容许有限 Reidemeister 数自同构的群，并证明了这些群满足有限扭曲 Burnside-Frobenius 定理猜想。
A \textit{$3$-net} of order $n$ is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size $n$, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the three classes. The current interest around $3$-nets (embedded) in a projective plane $PG(2,K)$, defined over a field $K$ of characteristic $p$, arose from algebraic geometry. It is not difficult to find $3$-nets in $PG(2,K)$ as far as $0<p\le n$. However, only a few infinite families of $3$-nets in $PG(2,K)$ are known to exist whe…
Suppose, $G$ is a residually finite group of finite upper rank admitting an automorphism $φ$ with finite Reidemeister number $R(φ)$ (the number of $φ$-twisted conjugacy classes). We prove that such $G$ is soluble-by-finite (in other words, any residually finite group of finite upper rank, which is not soluble-by-finite, has the $R_\infty$ property). This reduction is the first step in the proof of the second main theorem of the paper: suppose, $G$ is a residually finite group of finite Prüfer rank and $φ$ is its automorphism with $R(φ)<\infty$; then $R(φ)$ is equal to the number of equivalence…
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