3,197 results on '"Conjugacy class"'
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2. On classification of certain class of CP-groups.
- Author
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Foruzanfar, Zeinab and Rezaei, Mehdi
- Abstract
In this paper, we determine all finite groups in which each element has order a power of a prime and have at most five non-central conjugacy classes with representatives of orders multiples of p, for each arbitrary prime number p. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. روههايي كه مجموعهي اعضاي صفرشوي آنها اجتماع دقيقا سه كلاس تزويج است.
- Author
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سجاد محمود رباطي
- Abstract
Introduction Let G be a finite group and let Irr(G) be the set of irreducible characters of G. We say that an element g in G is a vanishing element if there exists some XE Irr(G) such that x(g) = 0. In this paper, we investigate the influence of the number of the columns containing some zeros in character table of G on the algebraic structure of G. Material and Methods Let Van(G) be the set of all vanishing elements, in other words, Van(G) = {g E G❘ 3χ € Irr(G) X(g) = 0 }. We can easily check that Van(G) is the union of some conjugacy classes. Burnside show that Van(G) = Ø if and only if G is an abelian group. We know that finite groups whose set of vanishing elements is the union of at most three conjugacy classes are solvable. Using this result, we provide a relatively short proof for the main theorem. Results and discussion In this paper, we classify finite groups whose set of vanishing elements is the union of exactly three conjugacy classes. Conclusion If the set of vanishing elements of a finite group G is the union of exactly three conjugacy classes then one of the following,situations:occurs Gis isomorphic to D8, Q8, or S4. G is a Frobenius group with abelian kernel of odd order and cyclic complement of order 4. GFxZ3, in which F is a Frobenius group with abelian kernel of odd order and complement of order 2. [ABSTRACT FROM AUTHOR]
- Published
- 2023
4. Designs from maximal subgroups and conjugacy classes of $\mathrm{PSL}(2,q)$, $q$ odd
- Author
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Xavier Mbaale, Bernardo Rodrigues, and Seiran Zandi
- Subjects
linear group ,design ,conjugacy class ,maximal subgroup ,primitive permutation representation ,Mathematics ,QA1-939 - Abstract
In this paper, using a method of construction of $1$-designs which are not necessarily symmetric, introduced by Key and Moori, we determine a number of $1$-designs with interesting parameters from the maximal subgroups and the conjugacy classes of elements of the group $PSL(2,q)$ for $q$ a power of an odd prime.
- Published
- 2023
- Full Text
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5. Conjugacy classes of left ideals of Sweedler's four-dimensional algebra $ H_{4} $
- Author
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Fengxia Gao and Jialei Chen
- Subjects
left ideal ,idempotent ,semigroup ,nilpotent left ideal ,conjugacy class ,Mathematics ,QA1-939 - Abstract
Let $ A $ be a finite-dimensional algebra with identity over the field $ \mathbb{F} $, $ U(A) $ be the group of units of $ A $ and $ L(A) $ be the set of left ideals of $ A $. It is well known that there is an equivalence relation $ \sim $ on $ L(A) $ by defining $ L_1\sim L_2\in L(A) $ if and only if there exists some $ u\in U(A) $ such that $ L_{1} = L_{2}u $. $ C(A) = \{[L]|L\in L(A)\} $ is the set of equivalence classes determined by the relation $ \sim $, which is a semigroup with respect to the natural operation $ [L_1][L_2] = [L_1L_2] $ for any $ L_1, L_2 \in L(A) $. The purpose of this paper is to describe the structures of semigroup of conjugacy classes of left ideals for the Sweedler's four-dimensional Hopf algebra $ H_{4} $.
- Published
- 2022
- Full Text
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6. Regularity of extended conjugate graphs of finite groups
- Author
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Piyapat Dangpat and Teerapong Suksumran
- Subjects
extended conjugate graph ,regular graph ,conjugacy class ,nilpotent group ,p-group ,Mathematics ,QA1-939 - Abstract
The extended conjugate graph associated to a finite group $ G $ is defined as an undirected graph with vertex set $ G $ such that two distinct vertices joined by an edge if they are conjugate. In this article, we show that several properties of finite groups can be expressed in terms of properties of their extended conjugate graphs. In particular, we show that there is a strong connection between a graph-theoretic property, namely regularity, and an algebraic property, namely nilpotency. We then give some sufficient conditions and necessary conditions for the non-central part of an extended conjugate graph to be regular. Finally, we study extended conjugate graphs associated to groups of order $ pq $, $ p^3 $, and $ p^4 $, where $ p $ and $ q $ are distinct primes.
- Published
- 2022
- Full Text
- View/download PDF
7. Non-solvable groups all of whose indices are odd-square-free.
- Author
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MAHMOOD ROBATI, Sajjad and HAFEZIEH BALAMAN, Roghayeh
- Subjects
- *
FINITE groups , *ODD numbers , *DIVISIBILITY groups , *CONJUGACY classes - Abstract
Given a finite group G and x ∈ G, the class size of x in G is called odd-square-free if it is not divisible by the square of any odd prime number. In this paper, we show that if G is a nonsolvable finite group, all of whose class sizes are odd-square-free, then we have some control on the structure of G, which is an answer to the dual of the question mentioned by Huppert in [5]. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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8. The Divisibility Graph for F-Groups.
- Author
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Khoshnevis, D. and Mostaghim, Z.
- Subjects
- *
CONJUGACY classes , *FINITE groups , *COMPLETE graphs , *DIVISIBILITY groups , *REGULAR graphs - Abstract
A graph is called the divisibility graph of if its vertex set is the set of noncentral conjugacy class sizes of and there is an edge between vertices and if and only if or . We determine the number of connected components of the divisibility graph when is an F-group. A finite group is called an F-group if for every , implies . We also prove that if the divisibility graph in which is an F-group is a -regular graph, then the divisibility graph is a complete graph with vertices. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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9. Conjugacy Class Sizes in Affine Semi-linear Groups
- Author
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Hossein Shahrtash
- Subjects
affine semi-linear group ,conjugacy class ,Mathematics ,QA1-939 - Abstract
The aim of this work is to study the structure and sizes of conjugacy classes in certain affine semi-linear groups. This provides a wealth of finite groups of small conjugate rank that are solvable and non-nilpotent.
- Published
- 2020
- Full Text
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10. Groups with many Subgroups which are Commensurable with some Normal Subgroup
- Author
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Ulderico Dardano and Silvana Rinauro
- Subjects
nearly normal subgroup ,core-finite subgroup ,minimal condition ,conjugacy class ,infinite rank ,Mathematics ,QA1-939 - Abstract
A subgroup H of a group G is called commensurable with a normal subgroup (cn) if there is N C G such that |HN/(H ∩ N)| is finite. We characterize generalized radical groups G which have one of the following finiteness conditions: (A) the minimal condition on non-cn subgroups of G; (B) the non-cn subgroups of G fall into finitely many conjugacy classes; (C) the non-cn subgroups of G have finite rank.
- Published
- 2019
- Full Text
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11. A New Approach to the Q-Conjugacy Character Tables of Finite Groups.
- Author
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Moghani, Ali
- Subjects
- *
FINITE groups , *HERMITIAN forms , *CHARACTER , *CONJUGACY classes - Abstract
In this paper, we study the Q-conjugacy character table of an arbitrary finite group and introduce a general relation between the degrees of Q-conjugacy characters with their corresponding reductions. This could be accomplished by using the Hermitian symmetric form. We provide a useful technique to calculate the character table of a finite group when its corresponding Qconjugacy character table is given. Then, we evaluate our results in some useful examples. Finally, by using GAP (Groups, Algorithms and Programming) package, we calculate all the dominant classes of the sporadic Conway group Co2 enabling us to find all possible the integer-valued characters for the Conway group Co2. [ABSTRACT FROM AUTHOR]
- Published
- 2020
12. Commuting involutions and elementary abelian subgroups of simple groups
- Author
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Geoffrey R. Robinson and Robert M. Guralnick
- Subjects
Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,Existential quantification ,010102 general mathematics ,Representation (systemics) ,Group Theory (math.GR) ,01 natural sciences ,20D06 (Primary_, 20C15 (Secondary) ,Mathematics::Group Theory ,Conjugacy class ,Simple group ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Abelian group ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that contains a member of each conjugacy class of involutions of G.
- Published
- 2022
13. p-Regular conjugacy classes and p-rational irreducible characters
- Author
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Attila Maróti and Nguyen Ngoc Hung
- Subjects
Finite group ,Algebra and Number Theory ,010102 general mathematics ,Field (mathematics) ,Group Theory (math.GR) ,20E45, 20C15, 20D05, 20D06, 20D10 ,01 natural sciences ,Upper and lower bounds ,Prime (order theory) ,Combinatorics ,Conjugacy class ,Simple group ,0103 physical sciences ,FOS: Mathematics ,Order (group theory) ,Rank (graph theory) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $G$ be a finite group of order divisible by a prime $p$. The number of $p$-regular and $p'$-regular conjugacy classes of $G$ is at least $2\sqrt{p-1}$. Also, the number of $p$-rational and $p'$-rational irreducible characters of $G$ is at least $2\sqrt{p-1}$. Along the way we prove a uniform lower bound for the number of $p$-regular classes in a finite simple group of Lie type in terms of its rank and size of the underlying field., 46 pages
- Published
- 2022
14. Normalized unit groups and their conjugacy classes.
- Author
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Kaur, S and Khan, M
- Subjects
CONJUGACY classes ,ABELIAN groups ,NONABELIAN groups ,FINITE fields ,GROUP rings - Abstract
Let G = H × A be a finite 2-group, where H is a non-abelian group of order 8 and A is an elementary abelian 2-group. We obtain a normal complement of G in the normalized unit group V(FG) and in the unitary subgroup V ∗ (F G) over the field F with 2 elements. Further, for a finite field F of characteristic 2, we derive class size of elements of V(FG). Moreover, we provide class representatives of V ∗ (F H) . [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
15. Algebras with finitely many conjugacy classes of left ideals versus algebras of finite representation type.
- Author
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Mȩcel, Arkadiusz and Okniński, Jan
- Subjects
- *
FINITE fields , *CONJUGACY classes , *IDEALS (Algebra) , *DIMENSIONS , *RING theory - Abstract
Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A -modules have dimension at least 6. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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16. The Cellular Automaton
- Author
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’t Hooft, Gerard, van Beijeren, Henk, Series editor, Blanchard, Philippe, Series editor, Busch, Paul, Series editor, Coecke, Bob, Series editor, Dieks, Dennis, Series editor, Dittrich, Bianca, Series editor, Dürr, Detlef, Series editor, Durrer, Ruth, Series editor, Frigg, Roman, Series editor, Fuchs, Christopher, Series editor, Ghirardi, Giancarlo, Series editor, Giulini, Domenico J. W., Series editor, Jaeger, Gregg, Series editor, Kiefer, Claus, Series editor, Landsman, Nicolaas P., Series editor, Maes, Christian, Series editor, Murao, Mio, Series editor, Nicolai, Hermann, Series editor, Petkov, Vesselin, Series editor, Ruetsche, Laura, Series editor, Sakellariadou, Mairi, Series editor, van der Merwe, Alwyn, Series editor, Verch, Rainer, Series editor, Werner, Reinhard, Series editor, Wüthrich, Christian, Series editor, Young, Lai-Sang, Series editor, and 't Hooft, Gerard
- Published
- 2016
- Full Text
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17. A parametrization of sheets of conjugacy classes in bad characteristic
- Author
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Filippo Ambrosio, Giovanna Carnovale, and Francesco Esposito
- Subjects
General Mathematics ,sheet ,FOS: Mathematics ,conjugacy class ,Group Theory (math.GR) ,Representation Theory (math.RT) ,Mathematics - Group Theory ,Mathematics - Representation Theory - Abstract
Let G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs $(M,{\mathcal O})$ where M is the identity component of the centralizer of a semisimple element in G and ${\mathcal O}$ is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.
- Published
- 2023
18. CLASS LENGTH OF ELEMENTS OF GROUP IN THE NORMALIZED UNIT GROUP.
- Author
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Kaur, S. and Khan, M.
- Subjects
CONJUGACY classes ,FINITE fields ,UNITARY groups ,GROUP rings - Abstract
Let F be a finite field of characteristic p > 0. In this article, we obtain a relation between the class length of elements of a finite p-group G in the normalized unit group V(FG) and its unitary subgroup V
* (FG), when p is an odd prime. We also provide the size of the conjugacy class of non-central elements of a group G in V(FG), where either G is any finite p-group with nilpotency class 2 or G is a p-group with nilpotency class 3 such that |G| ≤ p5 . [ABSTRACT FROM AUTHOR]- Published
- 2019
- Full Text
- View/download PDF
19. Explicit Artin maps into PGL2
- Author
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Antonia W. Bluher
- Subjects
Mathematics - Number Theory ,Group (mathematics) ,General Mathematics ,Order (ring theory) ,Unipotent ,Characterization (mathematics) ,Additive polynomial ,Combinatorics ,11R58, 11T30 ,Conjugacy class ,FOS: Mathematics ,Number Theory (math.NT) ,Galois extension ,Prime power ,Mathematics - Abstract
Let $G$ be a subgroup of ${\rm PGL}_2({\mathbb F}_q)$, where $q$ is any prime power, and let $Q \in {\mathbb F}_q[x]$ such that ${\mathbb F}_q(x)/{\mathbb F}_q(Q(x))$ is a Galois extension with group $G$. By explicitly computing the Artin map on unramified degree-1 primes in ${\mathbb F}_q(Q)$ for various groups $G$, interesting new results emerge about finite fields, additive polynomials, and conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. For example, by taking $G$ to be a unipotent group, one obtains a new characterization for when an additive polynomial splits completely over ${\mathbb F}_q$. When $G = {\rm PGL}_2({\mathbb F}_q)$, one obtains information about conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. When $G$ is the group of order 3 generated by $x \mapsto 1 - 1/x$, one obtains a natural tripartite symbol on ${\mathbb F}_q$ with values in ${\mathbb Z}/3{\mathbb Z}$. Some of these results generalize to ${\rm PGL}_2(K)$ for arbitrary fields $K$. Apart from the introduction, this article is written from first principles, with the aim to be accessible to graduate students or advanced undergraduates. An earlier draft of this article was published on the Math arXiv in June 2019 under the title {\it More structure theorems for finite fields}., Comment: Version 4 contains minor corrections and updates to the bibliograpy. Version 3 is a major revision, including a change in the title from "More structure theorems for finite fields" to "Explicit Artin maps into PGL2". The author thanks Xander Faber for insightful comments that led to the change in the title
- Published
- 2022
20. The Class Equation and the Commutativity Degree for Complete Hypergroups
- Author
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Andromeda Cristina Sonea and Irina Cristea
- Subjects
complete hypergroup ,commutativity degree ,conjugacy class ,class equation ,Mathematics ,QA1-939 - Abstract
The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation.
- Published
- 2020
- Full Text
- View/download PDF
21. Minimal reduction type and the Kazhdan–Lusztig map
- Author
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Zhiwei Yun
- Subjects
Classical group ,Pure mathematics ,Weyl group ,Fiber (mathematics) ,General Mathematics ,Type (model theory) ,Section (fiber bundle) ,Mathematics::Group Theory ,Nilpotent ,symbols.namesake ,Conjugacy class ,Mathematics::Quantum Algebra ,symbols ,Affine transformation ,Mathematics::Representation Theory ,Mathematics - Abstract
We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig in Lfromto, (2011) and that the Kazhdan–Lusztig map in Kazhdan and Lusztig, (1998) is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the “skeleta” of affine Springer fibers.
- Published
- 2021
22. A new family of Dai-Liao conjugate gradient methods with modified secant equation for unconstrained optimization
- Author
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Yutao Zheng
- Subjects
Conjugacy class ,Wolfe line search ,Conjugate gradient method ,Applied mathematics ,Unconstrained optimization ,Management Science and Operations Research ,Computer Science Applications ,Theoretical Computer Science ,Mathematics - Abstract
In this paper, a new family of Dai-Liao–type conjugate gradient methods are proposed for unconstrained optimization problem. In the new methods, the modified secant equation used in [H. Yabe and M. Takano, Comput. Optim. Appl. 28 (2004) 203–225] is considered in Dai and Liao’s conjugacy condition. Under some certain assumptions, we show that our methods are globally convergent for general functions with strong Wolfe line search. Numerical results illustrate that our proposed methods can outperform some existing ones.
- Published
- 2021
23. Generic properties of homeomorphisms preserving a given dynamical simplex
- Author
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Julien Melleray
- Subjects
Mathematics::Dynamical Systems ,Simplex ,Generic property ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Cantor space ,Dynamical Systems (math.DS) ,Combinatorics ,Mathematics::Group Theory ,Conjugacy class ,Clopen set ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Invariant measure ,Mathematics - Dynamical Systems ,Element (category theory) ,Invariant (mathematics) ,Mathematics - Abstract
Given a dynamical simplex $K$ on a Cantor space $X$, we consider the set $G_K^*$ of all homeomorphisms of $X$ which preserve all elements of $K$ and have no nontrivial clopen invariant subset. Generalising a theorem of Yingst, we prove that for a generic element $g$ of $G_K^*$ the set of invariant measures of $g$ is equal to $K$. We also investigate when there exists a generic conjugacy class in $G_K^*$ and prove that this happens exactly when $K$ has only one element, which is the unique invariant measure associated to some odometer; and that in that case the conjugacy class of this odometer is generic in $G_K^*$., Updated (and final) version
- Published
- 2021
24. The transitive groups of degree 48 and some applications
- Author
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Derek F. Holt, Gareth Tracey, and Gordon F. Royle
- Subjects
Transitive relation ,Algebra and Number Theory ,Degree (graph theory) ,Cayley graph ,010102 general mathematics ,Magma (algebra) ,Permutation group ,01 natural sciences ,Combinatorics ,Conjugacy class ,Symmetric group ,0103 physical sciences ,Enumeration ,010307 mathematical physics ,0101 mathematics ,QA ,Mathematics - Abstract
The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the 195 826 352 conjugacy classes of transitive subgroups of the symmetric group S 48 of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive groups of degree 48, of which there are 1 538 868 366 , all but 0.1625% of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published as a separate paper.
- Published
- 2022
25. A class of dependent Dirichlet processes via latent multinomial processes
- Author
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Luis E. Nieto-Barajas
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Class (set theory) ,Series (mathematics) ,Context (language use) ,Dirichlet distribution ,Methodology (stat.ME) ,Set (abstract data type) ,Dirichlet process ,symbols.namesake ,Conjugacy class ,symbols ,Applied mathematics ,Multinomial distribution ,Statistics, Probability and Uncertainty ,Statistics - Methodology ,Mathematics - Abstract
We describe a procedure to introduce general dependence structures on a set of Dirichlet processes. Dependence can be in one direction to define a time series or in two directions to define spatial dependencies. More directions can also be considered. Dependence is induced via a set of latent processes and exploit the conjugacy property between the Dirichlet and the multinomial processes to ensure that the marginal law for each element of the set is a Dirichlet process. Dependence is characterised through the correlation between any two elements. Posterior distributions are obtained when we use the set of Dirichlet processes as prior distributions in a bayesian nonparametric context. Posterior predictive distributions induce partially exchangeable sequences defined by generalised P\'olya urs. A numerical example to illustrate is also included.
- Published
- 2021
26. Groups whose prime graph on class sizes has a cut vertex
- Author
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Silvio Dolfi, Lucia Sanus, Emanuele Pacifici, and Víctor Sotomayor
- Subjects
Class (set theory) ,Finite group ,General Mathematics ,Prime number ,Group Theory (math.GR) ,Vertex (geometry) ,Set (abstract data type) ,Combinatorics ,Conjugacy class ,Simple (abstract algebra) ,Prime graph ,FOS: Mathematics ,20E45 ,Finite groups, Conjugacy classes, Prime graph ,Mathematics - Group Theory ,Mathematics - Abstract
Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two vertices $p$ and $q$ being adjacent if and only if $pq$ divides some conjugacy class size of $G$. In the present paper, we classify the finite groups $G$ for which $\Delta(G)$ has a cut vertex.
- Published
- 2021
27. Enumerating partial linear transformations in a similarity class
- Author
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Samrith Ram and Akansha Arora
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Similarity (geometry) ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Linear subspace ,Square matrix ,Combinatorics ,Linear map ,Finite field ,Conjugacy class ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,05A05, 05A10, 15B33 ,Subspace topology ,Mathematics ,Vector space - Abstract
Let $V$ be a finite-dimensional vector space over the finite field ${\mathbb F}_q$ and suppose $W$ and $\widetilde{W}$ are subspaces of $V$. Two linear transformations $T:W\to V$ and $\widetilde{T}:\widetilde{W}\to V$ are said to be similar if there exists a linear isomorphism $S:V\to V$ with $SW=\widetilde{W}$ such that $S\circ T=\widetilde{T}\circ S $. Given a linear map $T$ defined on a subspace $W$ of $V$, we give an explicit formula for the number of linear maps that are similar to $T$. Our results extend a theorem of Philip Hall that settles the case $W=V$ where the above problem is equivalent to counting the number of square matrices over ${\mathbb F}_q$ in a conjugacy class., 15 pages, 3 figures
- Published
- 2021
28. Conjugacy of yield and its structural elements in spring soft wheat samples
- Author
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Irina F. Demina
- Subjects
grain weight per ear ,triticum aestivum l ,Yield (engineering) ,correlation relation ,Agriculture ,Spring (mathematics) ,yield ,the elements of the yield structure ,coefficient of variation ,Conjugacy class ,Agronomy ,General Materials Science ,Mathematics - Abstract
The article presents an analysis of genotypic correlations between the yield of 33 variety samples of spring soft wheat and elements of its structure in the conditions of Penza region, the degree of variability of agronomic valuable traits during the years of research (2018-2020) has been determined. It has been established, that the low-varying agronomic valuable traits (CV = 7.8-9.9 %) include the wheat ear length, number of spikelets in the ear, weight of 1000 grains; moderately varying traits (СV = 13.8-15.6 %) include productive bushiness capacity, the number of grains in the ear and weight of grains in one ear; highly-varying traits (СV = 21.7-22.7 %) include the number of grains per ear and weight of the grain per ear. A strong positive interrelation has been established between the yield of spring soft wheat and the number of grains per ear (r = 0.706...0.816) and weight of grain per ear (r = 0.754...0.875). There has been revealed an average positive interrelation between the yield and the weight of ears (r = 0.467...0.621), the number of spikelets per ear (r = 0.358...0.582), the number of grains per plant (r = 0.446...0.541) and the weight of grain per plant (r = 0.309...0.608). The correlation dependence of yield on productive bushiness (r = 0.091…0.415), ear length (r = 0.074…0.503) and weight of 1000 grains (r = 0.193…0.583) turned out to be unstable. Thus, the formation of grain yield was influenced by the number of grains per ear and the weight of grain per ear. The analysis showed the degree of influence of various elements of productivity on the formation of yield of spring soft wheat variety samples that provides a more targeted selection in the breeding process.
- Published
- 2021
29. Unitary conjugacy for type III subfactors and W$^*$-superrigidity
- Author
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Yusuke Isono
- Subjects
46L36 ,Pure mathematics ,46L10 ,37A20 ,Mathematics::Operator Algebras ,46L55 ,Applied Mathematics ,General Mathematics ,Mathematics - Operator Algebras ,Tomita–Takesaki theory ,Characterization (mathematics) ,Type (model theory) ,W*-superrigidity ,Unitary state ,Group action ,Conjugacy class ,Free product ,FOS: Mathematics ,Popa’s intertwining theory ,Element (category theory) ,Operator Algebras (math.OA) ,strong solidity ,Mathematics - Abstract
Let $A,B\subset M$ be inclusions of $\sigma$-finite von Neumann algebras such that $A$ and $B$ are images of faithful normal conditional expectations. In this article, we investigate Popa's intertwining condition $A\preceq_MB$ using their modular actions. In the main theorem, we prove that if $A\preceq_MB$ holds, then an intertwining element for $A\preceq_MB$ also intertwines some modular flows of $A$ and $B$. As a result, we deduce a new characterization of $A\preceq_MB$ in terms of their continuous cores. Using this new characterization, we prove the first W$^*$-superrigidity type result for group actions on amenable factors. As another application, we characterize stable strong solidity for free product factors in terms of their free product components., Comment: 35 pages
- Published
- 2021
30. The local-to-global property for Morse quasi-geodesics
- Author
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Hung Cong Tran, Davide Spriano, and Jacob Russell
- Subjects
Computer Science::Machine Learning ,Pure mathematics ,Geodesic ,General Mathematics ,Regular polygon ,Geometric Topology (math.GT) ,Group Theory (math.GR) ,Type (model theory) ,Morse code ,Translation (geometry) ,Computer Science::Digital Libraries ,Mapping class group ,law.invention ,Mathematics::Group Theory ,Mathematics - Geometric Topology ,Statistics::Machine Learning ,Quasiconvex function ,Conjugacy class ,law ,FOS: Mathematics ,Computer Science::Mathematical Software ,20F65, 20F67 ,Mathematics - Group Theory ,Mathematics - Abstract
We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan-Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska., Mathematische Zeitschrift, 300 (2)
- Published
- 2021
31. Invariants Of Uniform Conjugacy On Uniform Dynamical System
- Author
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Ihsan Jabbar Khadim and Alaa Saeed Abboud
- Subjects
Pure mathematics ,Conjugacy class ,Complementary and alternative medicine ,Pharmaceutical Science ,Pharmacology (medical) ,Uniform space ,Dynamical system (definition) ,Expansive ,Homeomorphism ,Generator (mathematics) ,Mathematics - Abstract
In this paper, we present some important dynamical concepts on uniform space such as the uniform minimal systems, uniform shadowing, and strong uniform shadowing. We explain some definitions and theorems such as definition uniform expansive, weak uniform expansive, uniform generator, and the proof of the theorems for them. We prove that if be a homeomorphism on a compact uniform space then has uniform shadowing if and only if has uniform shadowing, so if has strong uniform shadowing if and only if has strong uniform shadowing. We also show that and be two uniform homeomorphisms on compact uniform spaces and , if is a uniform conjugacy from to , then . Besides some other results.
- Published
- 2021
32. Several Results on Conjugacy Class Sizes of Some Elements of Finite Groups
- Author
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Yongcai Ren
- Subjects
Mathematics::Group Theory ,Pure mathematics ,Finite group ,Conjugacy class ,Simple (abstract algebra) ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Element (category theory) ,01 natural sciences ,Mathematics - Abstract
Let G be a finite group. For an element x of G, xG denotes the conjugacy class of x in G. |xG| is called the size of the conjugacy class xG. In this paper, we establish several results on conjugacy class sizes of some elements of finite groups. In addition, we give a simple and clearer proof of a known result.
- Published
- 2021
33. A note on groups with many locally supersoluble subgroups
- Author
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Francesco de Giovanni and Marco Trombetti
- Subjects
locally supersoluble group ,minimal condition ,conjugacy class ,Mathematics ,QA1-939 - Abstract
It is proved here that if G is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then G is either locally supersoluble or a \vCernikov group. The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. As a consequence, it is shown that any infinite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble.
- Published
- 2015
34. COUNTABLY RECOGNIZABLE CLASSES OF GROUPS WITH RESTRICTED CONJUGACY CLASSES.
- Author
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DE GIOVANNI, FRANCESCO and TROMBETTI, MARCO
- Subjects
- *
CONJUGACY classes , *MAXIMAL subgroups - Abstract
A group class X is said to be countably recognizable if a group belongs to X whenever all its countable subgroups lie in X. It is proved here that most of the relevant classes of groups defined by restrictions on the conjugacy classes are countably recognizable. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
35. Equality of orders of a set of integers modulo a prime
- Author
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Olli Järviniemi
- Subjects
Reduction (recursion theory) ,Mathematics - Number Theory ,Generalization ,Applied Mathematics ,General Mathematics ,Modulo ,Prime (order theory) ,Combinatorics ,Riemann hypothesis ,symbols.namesake ,Conjugacy class ,FOS: Mathematics ,symbols ,Computer Science::Symbolic Computation ,Number Theory (math.NT) ,Galois extension ,Finitely-generated abelian group ,Mathematics - Abstract
For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb{Q}^{\times}$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb{Q}$ and a union of conjugacy classes $C \subset \text{Gal}(F/\mathbb{Q})$, we develop methods for determining if there exists infinitely many primes $p$ such that the index of the reduction of $W_i$ modulo $p$ divides $k_i$ and such that the Artin symbol of $p$ on $F$ is contained in $C$. The results are a multivariable generalization of H.W. Lenstra's work. As an application, we determine all integers $a_1, \ldots , a_n$ such that $\text{ord}_p(a_1) = \ldots = \text{ord}_p(a_n)$ for infinitely many primes $p$. We also discuss the set of those $p$ for which $\text{ord}_p(a_1) > \ldots > \text{ord}_p(a_n)$. The obtained results are conditional to a generalization of the Riemann hypothesis., 20 pages. Add section on Kummer-type extensions and improve exposition
- Published
- 2021
36. Linear Groups with Restricted Conjugacy Classes
- Author
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Marco Trombetti, B. A. F. Wehrfritz, F. de Giovanni, de Giovanni, F., Trombetti, M., and Wehrfritz, B. A. F.
- Subjects
Class (set theory) ,Layer ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,01 natural sciences ,Conjugacy cla ,010305 fluids & plasmas ,Combinatorics ,Linear group ,Conjugacy class ,0103 physical sciences ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
In this paper we characterize, in terms of their conjugacy classes, linear groups G such that $$G/\zeta _k(G)$$ G / ζ k ( G ) belongs to a certain group class $$\mathfrak {X}$$ X for several natural choices of $$\mathfrak {X}$$ X . Moreover, a description is given of linear groups with restrictions on layers.
- Published
- 2021
37. Equivariant -absorption theorem for exact groups
- Author
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Yuhei Suzuki
- Subjects
Pure mathematics ,Algebra and Number Theory ,Conjugacy class ,Group (mathematics) ,Countable set ,Equivariant map ,Absorption (electromagnetic radiation) ,Mathematics - Abstract
We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal {O}_{2}$-absorbing, pointwise outer action on the Cuntz algebra $\mathcal {O}_{2}$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal {O}_{2}$-absorption theorem for these groups.
- Published
- 2021
38. A Note on the Conjugacy Between Two Critical Circle Maps
- Author
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Utkir A. Safarov
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Conjugacy class ,Singular function ,Critical point (thermodynamics) ,General Mathematics ,General Physics and Astronomy ,Rotation number ,Mathematics - Abstract
We study a conjugacy between two critical circle homeomorphisms with irrational rotation number. Let fi, i = 1, 2 be a C3 circle homeomorphisms with critical point x(i) cr of the order 2mi + 1. We prove that if 2m1 + 1 ̸= 2m2 + 1, then conjugating between f1 and f2 is a singular function. Keywords: circle homeomorphism, critical point, conjugating map, rotation number, singular function
- Published
- 2021
39. INTEGRABILITY OF CLASSICAL AFFINE W-ALGEBRAS
- Author
-
Mamuka Jibladze, Alberto De Sole, Daniele Valeri, and Victor G. Kac
- Subjects
Pure mathematics ,Integrable system ,FOS: Physical sciences ,Mathematics::Group Theory ,Conjugacy class ,Simple (abstract algebra) ,Lie algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematical Physics ,Mathematics ,W-algebras ,integrable systems ,generalized Drinfeld-Sokolov hierarchies ,Algebra and Number Theory ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Hierarchy (mathematics) ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Mathematical Physics (math-ph) ,Nilpotent ,Rings and Algebras (math.RA) ,Geometry and Topology ,Affine transformation ,Exactly Solvable and Integrable Systems (nlin.SI) ,Element (category theory) ,Mathematics - Representation Theory - Abstract
We prove that all classical affine W-algebras W(g,f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G_2, one in F_4, and five in E_8., Comment: 18 pages
- Published
- 2021
40. Combinatorial, piecewise-linear, and birational homomesy for products of two chains
- Author
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David M. Einstein and James Propp
- Subjects
Dense set ,010102 general mathematics ,Order (ring theory) ,Polytope ,0102 computer and information sciences ,01 natural sciences ,Noncommutative geometry ,Orthant ,Combinatorics ,Conjugacy class ,010201 computation theory & mathematics ,Product (mathematics) ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Partially ordered set ,Mathematics - Abstract
This article illustrates the dynamical concept of $homomesy$ in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and shows the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley, and then lifted to birational operations on the positive orthant in $\mathbb{R}^{|P|}$ and indeed to a dense subset of $\mathbb{C}^{|P|}$. When the poset $P$ is a product of a chain of length $a$ and a chain of length $b$, these lifted operations have order $a+b$, and exhibit the homomesy phenomenon: the time-averages of various quantities are the same in all orbits. One important tool is a concrete realization of the conjugacy between rowmotion and promotion found by Striker and Williams; this $recombination$ $map$ allows us to use homomesy for promotion to deduce homomesy for rowmotion. NOTE: An earlier draft showed that Stanley's transfer map between the order polytope and the chain polytope arises as the tropicalization of an analogous map in the bilinear realm; in 2020 we removed this material for the sake of brevity, especially after Joseph and Roby generalized our proof to the noncommutative realm (see arXiv:1909.09658v3). Readers who nonetheless wish to see our proof can find the September 2018 draft of this preprint through the arXiv.
- Published
- 2021
41. Advanced Group Theory
- Author
-
Knapp, Anthony W. and Knapp, Anthony W.
- Published
- 2006
- Full Text
- View/download PDF
42. Groups and Group Actions
- Author
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Knapp, Anthony W. and Knapp, Anthony W.
- Published
- 2006
- Full Text
- View/download PDF
43. Non-nilpotent groups with three conjugacy class of non-normal subgroups
- Author
-
Hamid Mousavi
- Subjects
Non-Normal Subgroup ,Conjugacy Class ,Non-Nilpotent Group ,Mathematics ,QA1-939 - Abstract
For a finite group $G$ let $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$. The aim of this paper is to classify all the non-nilpotent groups with $nu(G)=3$.
- Published
- 2014
44. On properties of translation groups in the affine general linear group with applications to cryptography
- Author
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Massimiliano Sala, Marco Calderini, and Roberto Civino
- Subjects
Algebra and Number Theory ,business.industry ,010102 general mathematics ,20B35, 15A21, 94A60 ,General linear group ,Cryptography ,Group Theory (math.GR) ,Translation (geometry) ,01 natural sciences ,Algebra ,Conjugacy class ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Abelian group ,business ,Representation (mathematics) ,Mathematics - Group Theory ,Computer Science::Cryptography and Security ,Vector space ,Mathematics - Abstract
The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Several combinatorial counting formulas and a classification of their conjugacy classes are given as well., Comment: to appear in Journal of Algebra
- Published
- 2021
45. On Diameter of Subgraphs of Commuting Graph in Symplectic Group for Elements of Order Three
- Author
-
Suzila Mohd Kasim and Athirah Nawawi
- Subjects
Combinatorics ,Finite group ,Multidisciplinary ,Symplectic group ,Conjugacy class ,Simple (abstract algebra) ,Graph (abstract data type) ,Order (group theory) ,Vertex (geometry) ,Symplectic geometry ,Mathematics - Abstract
Suppose G be a finite group and X be a subset of G. The commuting graph, denoted by C(G,X), is a simple undirected graph, where X ⊂G being the set of vertex and two distinct vertices x,y∈X are joined by an edge if and only if xy = yx. The aim of this paper was to describe the structure of disconnected commuting graph by considering a symplectic group and a conjugacy class of elements of order three. The main work was to discover the disc structure and the diameter of the subgraph as well as the suborbits of symplectic groups S4(2)', S4(3) and S6(2). Additionally, two mathematical formulas are derived and proved, one gives the number of subgraphs based on the size of each subgraph and the size of the conjugacy class, whilst the other one gives the size of disc relying on the number and size of suborbits in each disc.
- Published
- 2021
46. Complete Classification of Degree 7 for Genus 1
- Author
-
Peshawa M. Khudhur
- Subjects
General Computer Science ,Degree (graph theory) ,Mathematics::Complex Variables ,General Chemistry ,General Biochemistry, Genetics and Molecular Biology ,Combinatorics ,Conjugacy class ,Symmetric group ,Genus (mathematics) ,Bijection ,Cover (algebra) ,Compact Riemann surface ,Mathematics ,Meromorphic function - Abstract
Assume that is a meromorphic fuction of degree n where X is compact Riemann surface of genus g. The meromorphic function gives a branched cover of the compact Riemann surface X. Classes of such covers are in one to one correspondence with conjugacy classes of r-tuples ( of permutations in the symmetric group , in which and s generate a transitive subgroup G of This work is a contribution to the classification of all primitive groups of degree 7, where X is of genus one.
- Published
- 2021
47. Representations of Motion Groups of Links via Dimension Reduction of TQFTs
- Author
-
Zhenghan Wang and Yang Qiu
- Subjects
Physics ,Quantum Physics ,Conjecture ,Topological quantum field theory ,010102 general mathematics ,Braid group ,FOS: Physical sciences ,Motion (geometry) ,Geometric Topology (math.GT) ,Statistical and Nonlinear Physics ,Torus ,State (functional analysis) ,01 natural sciences ,Centralizer and normalizer ,Combinatorics ,Mathematics - Geometric Topology ,Conjugacy class ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,010307 mathematical physics ,0101 mathematics ,Quantum Physics (quant-ph) ,Mathematical Physics - Abstract
Motion groups of links in the three sphere $\mathbb{S}^3$ are generalizations of the braid groups, which are motion groups of points in the disk $\mathbb{D}^2$. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each $1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in $\mathbb{S}^3$ with generalized axes from Dijkgraaf-Witten (DW) TQFTs inspired by dimension reduction. A succinct way to state our result is as a step toward a twisted generalization (Conjecture \ref{mainconjecture}) of a conjecture for DW theories of dimension reduction from $(3+1)$ to $(2+1)$: $\textrm{DW}^{3+1}_G \cong \oplus_{[g]\in [G]} \textrm{DW}^{2+1}_{C(g)}$, where the sum runs over all conjugacy classes $[g]\in [G]$ of $G$ and $C(g)$ the centralizer of any element $g\in [g]$. We prove a version of Conjecture \ref{mainconjecture} for the mapping class groups of closed manifolds and the case of torus links labeled by pure fluxes., Clarify the main conjecture as a twisted version of dimension reduction. To appear in Comm. Math Phys
- Published
- 2021
48. Computational complexity of k-block conjugacy
- Author
-
Rafael M. Frongillo and Tyler Schrock
- Subjects
Discrete mathematics ,Vertex (graph theory) ,Mathematics::Group Theory ,Mathematics::Dynamical Systems ,Conjugacy class ,General Computer Science ,Computational complexity theory ,Block (permutation group theory) ,Computational problem ,Subshift of finite type ,Representation (mathematics) ,Theoretical Computer Science ,Mathematics - Abstract
We consider several computational problems related to conjugacy between subshifts of finite type, restricted to k-block codes: verifying a proposed k-block conjugacy, deciding if two shifts admit a k-block conjugacy, and reducing the representation size of a shift via a k-block conjugacy. We give a polynomial-time algorithm for verification, and show GI - and NP -hardness for deciding conjugacy and reducing representation size, respectively. Our approach focuses on 1-block conjugacies between vertex shifts, from which we generalize to k-block conjugacies and to edge shifts. We conclude with several open problems.
- Published
- 2021
49. LOSIK CLASSES FOR CODIMENSION-ONE FOLIATIONS
- Author
-
Yaroslav V. Bazaikin and Anton S. Galaev
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Dynamical Systems ,Dynamical systems theory ,General Mathematics ,Holonomy ,Dynamical Systems (math.DS) ,Codimension ,Triviality ,Cohomology ,Characteristic class ,Conjugacy class ,Differential Geometry (math.DG) ,Mathematics::K-Theory and Homology ,FOS: Mathematics ,Ergodic theory ,Mathematics::Differential Geometry ,Mathematics - Dynamical Systems ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
Following Losik's approach to Gelfand's formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for non-triviality in terms of dynamical properties of generators of the holonomy groups are found. The non-triviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class, e.g, the Mizutani-Morita-Tsuboi Theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections., The final version accepted to Journal of the Institute of Mathematics of Jussieu
- Published
- 2021
50. Local Lyapunov spectrum rigidity of nilmanifold automorphisms
- Author
-
Jonathan DeWitt
- Subjects
Mathematics::Dynamical Systems ,Rigidity (psychology) ,Dynamical Systems (math.DS) ,Lyapunov exponent ,01 natural sciences ,Combinatorics ,symbols.namesake ,Conjugacy class ,Computer Science::Systems and Control ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Nilmanifold ,Mathematics ,Algebra and Number Theory ,Computer Science::Information Retrieval ,Applied Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Automorphism ,Nonlinear Sciences::Chaotic Dynamics ,symbols ,Irreducibility ,010307 mathematical physics ,Analysis - Abstract
We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/\Gamma$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$ whenever $f$ is $C^{1+}$ and has the same volume Lyapunov spectrum as $L$. For $L$ with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to $L$ satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents., Comment: 40 pages; comments are welcome
- Published
- 2021
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