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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.

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

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

If C is a class of groups, a C -injector of a finite group G is a subgroup V of G with the property that V ∩ K is a C -maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschutz and B. Hartley states the existence and conjugacy of F -injectors in finite soluble groups for Fitting classes F . We shall show that for groups of nilpotent length at most 4, F -injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.

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.

In this article, a possible supercharacter theory based on the degrees of the irreducible characters of a finite group is introduced. An investigation is then included into which finite groups allow this particular supercharacter theory. In particular, it is shown that groups with at most seven conjugacy classes must allow this supercharacter theory. However, there are groups with eight conjugacy classes that do not have this supercharacter theory. In addition, it is shown that semidirect products $$A \rtimes B$$ where both A and B are cyclic groups and at least one is of prime order always allow this type of supercharacter theory.

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.

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

Let G be a finite group and V(G) be the set of all non-central conjugacy classes of G. Consider the conjugacy class graph Γ(G) of G which is defined as follows: its vertex set is the set V(G) and t...

The article is about the representation theory of an inner form G of a general linear group over a non-Archimedean local field. We introduce semisimple characters for G whose intertwining classes describe conjecturally via the Local Langlands correspondence the behaviour on wild inertia. These characters also play a potential role to understand the classification of irreducible smooth representations of inner forms of classical groups. We prove the intertwining formula for semisimple characters and an intertwining implies conjugacy like theorem. Further we show that endo-parameters for G, i.e. invariants consisting of simple endo-classes and a numerical part, classify the intertwining classes of semisimple characters for G. They should be the counter part for restrictions of Langlands-parameters to wild inertia under the Local Langlands correspondence.

Limited-memory variable metric methods based on the well-known Broyden-Fletcher-Goldfarb-Shanno (BFGS) update are widely used for large scale optimization. The block version of this update, derived for general objective functions in Vlcek and Luksan (Numerical Algorithms 2019), satisfies the secant conditions with all used difference vectors and for quadratic objective functions gives the best improvement of convergence in some sense, but the corresponding direction vectors are not descent directions generally. To guarantee the descent property of direction vectors and simultaneously violate the secant conditions as little as possible in some sense, two methods based on the block BFGS update are proposed. They can be advantageously used together with methods based on vector corrections for conjugacy. Here we combine two types of these corrections to satisfy the secant conditions with both the corrected and uncorrected (original) latest difference vectors. Global convergence of the proposed algorithm is established for convex and sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new methods.

Let [Formula: see text] be a prime and let [Formula: see text] be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of [Formula: see text] up to conjugacy. That is, we give a complete and irredundant list of [Formula: see text]-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of [Formula: see text] is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For [Formula: see text], we classify all finite irreducible subgroups of [Formula: see text]. Our classifications are available publicly in MAGMA.

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.

This paper defines Artin groups with m-gon structure and proves that for m > 3 in this class the problems of equality and conjugacy of words are solvable.

We give a positive answer to the problem of power conjugacy words in a tree product of free groups with cyclic amalgamation. It is a generalization of the well-known result obtained by S. Lipschutz for a free product of two free groups with cyclic amalgamation.

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.

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

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.

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

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)

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.

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.

The simple symplectic group Sp(8, 2) has 11 conugacy classes of maximal subgroups. The fourth maximal subgroup of Sp(8, 2) is a group of the form $$2^{10}{:}A_{8}:= \overline{G}.$$ In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford–Fischer Theory. We determined the inertia factor groups of $$\overline{G}$$ and there are 7 such groups having the forms: $$H_{1} = A_{8},$$ $$H_{2} = 2^{3}{:}GL(3,2),$$ $$H_{3} = 2^{4}{:}(S_{3}\times S_{3}),$$ $$H_{4} = 2^{3}{:}S_{4},$$ $$H_{5} = S_{5},$$ $$H_{6} = (S_{3}\times S_{3}){:}2$$ and $$H_{7} = 2 \times S_{4}.$$ The character table of $$\overline{G}$$ is a $$81 \times 81$$ complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 16.

Motivated by a 2004 conjecture by the author and J. Sangroniz, Y. Yang has recently proved that if G is solvable then the index in G of the 8th term of the ascending Fitting series is bounded in terms of the largest number of zeros in a row in the character table of G. In this note, we prove this result for arbitrary finite groups and propose a stronger form of the 2004 conjecture. We conclude the paper showing some possible ways to prove this strengthened conjecture.

There are many results that give information on the structure of a finite group in terms of properties that refer to its character degrees (or class sizes or element orders) and at most two primes. In this note, we try to extend some of these results considering three primes.

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

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.

In this paper we prove that the time part of the germ of an analytic family of vector fields with a Hopf bifurcation is rigid in the parameter. Time parts are associated with the temporal invariant of the analytic classification. Because the eigenvalues at zero are complex conjugate, time parts usually unfold in the hyperbolic direction, where the singular points are linearizable. We first identify the time part of a generic conformal family and prove that any weak holomorphic conjugacy between two time parts yields a biholomorphism analytic in the parameter. The existence of Fatou coordinates in both the Siegel and in the Poincare domains plays a fundamental role in the proof of this result.

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.

We previously have developed two methods for constructing codes and designs from finite simple groups. In a recent paper we introduced a new method (Method 3) for constructing codes and designs fro...

We establish a general duality theorem in a generalized conjugacy framework, which generalizes a classical result on the minimization of a convex function over a closed convex cone. Our theorem yields two quasiconvex duality schemes; one of them is of the surrogate duality type and is applicable to problems having an evenly quasiconvex objective function, whereas the other one is applicable to problems with Lipschitz quasiconvex objective functions and yields duals whose objective functions do not involve any surrogate constraint.

The article explores approaches to content of classification and principles of its construction. It is proposed to define the classification of other measures of a criminal legal nature as a slender, logically consistent set of individual categories (groups), including the distribution of other measures of a criminal legal nature, based on independent and subordinate classification grounds. The authors attempt to develop scientifically based grounds for classifying other criminal legal measures on the purpose and alternative (conjugation) of punishment. Other criminal legal measures applicable for the purpose of testing, raising or curing a subject, i. e. correcting their behavior or state of health, as well as other criminal legal measures aimed at restricting the property rights of the subject, affecting their property (including financial status) in order to restore social justice and compensation for damage caused to society and the State as a result of the commission of a crime, are identified.

Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

We prove the p-version of a conjecture raised by Isaacs et al. in Proc. AMS [8] on the degree of a primitive character that divides the size of some conjugacy class of a finite solvable group, thus generalizing Marchi's results [10] in J. Algebra 2020.

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

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

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.

In this paper, a new conjugacy condition is established to solve unconstrained optimization problems based on a new quasi-Newton equation. We present a modified Dai–Liao conjugate gradient method to solve unconstrained optimization problems with a new value of the parameter t based on the new conjugacy condition. The presented algorithm has the following properties: (i) the modified Dai–Liao conjugate gradient method considers both the gradient and function value information. (ii) The global convergence is achieved for the modified Dai–Liao conjugate gradient method under some suitable assumptions. (iii) Numerical experiments on unconstrained optimization problems and image restoration problems are conducted, and the numerical results show that our method is efficient.

We will show a centrally free action of an amenable rigid $$\hbox {C}^*$$ -tensor category on a properly infinite von Neumann algebra has the Rohlin property. Our main result is the classification of centrally free cocycle actions of an amenable rigid $$\hbox {C}^*$$ -tensor category up to approximate inner conjugacy on properly infinite von Neumann algebras. This is regarded as a generalization of classification of amenable discrete groups due to A. Connes, V. Jones and A. Ocneanu. We have the following two applications: a classification of centrally free actions of amenable discrete quantum groups of Kac type on von Neumann algebras and another proof of S. Popa’s celebrated classification result of amenable subfactors. As another application of the Rohlin property, we will prove the fullness of the crossed product of a full factor by a minimal action of a compact group.

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

Let $ {{\mathfrak{X}}} $ be a class of finite groups containing a group of even order and closed under subgroups, homomorphic images, and extensions. Then each finite group possesses a maximal $ {{\mathfrak{X}}} $ -subgroup of odd index and the study of the subgroups can be reduced to the study of the so-called submaximal $ {{\mathfrak{X}}} $ -subgroups of odd index in simple groups. We prove a theorem that deduces the description of submaximal $ {{\mathfrak{X}}} $ -subgroups of odd index in an alternating group from the description of maximal $ {{\mathfrak{X}}} $ -subgroups of odd index in the corresponding symmetric group. In consequence, we classify the submaximal soluble subgroups of odd index in alternating groups up to conjugacy.

We obtain some fairly general conditions on the linking numbers and geometric properties of a link, under which it has infinitely many conjugacy classes of [Formula: see text]-braid representatives if and only if it has one admitting an exchange move. We investigate a symmetry pattern of indices of conjugate iterated exchanged braids. We then develop a test based on the Burau matrix showing examples of knots admitting no minimal exchangeable braids, admitting non-minimal non-exchangeable braids, and admitting both minimal exchangeable and minimal non-exchangeable braids. This in particular proves that conjugacy, exchange moves and destabilization do not suffice to simplify braid representatives of a general link.

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.

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.

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

Given a finite group G, two elements are ≡m-related if they lie in exactly the same maximal subgroups of G. This equivalence relation was introduced by P. J. Cameron, A. Lucchini and C. M. Roney-Do...

Affine equivalent classes of Boolean functions have many applications in modern cryptography and circuit design. Previous publications have shown that affine equivalence on the entire space of Boolean functions can be computed up to 10 variables, but not on the quotient Boolean function space modulo functions of different degrees. Computing the number of equivalent classes of cosets of Reed-Muller code $\mathcal {R}(1,n)$ is equivalent to classifying Boolean functions modulo linear functions, which can be computed only when n ≤ 7. Based on the linear representation of the affine group $\mathcal {A}\mathcal {G}{\mathscr{L}}(n,2)$ on the quotient space $\mathcal {R}(s,n)/\mathcal {R}(k,n)$ , we obtain a useful counting formula to compute the number of equivalent classes. Instead of computing the conjugacy classes and representatives directly in $\mathcal {A}\mathcal {G}{\mathscr{L}}(n,2)$ , we reduce the computation complexity by introducing an isomorphic permutation group Pn and performing the computation in Pn. With the proposed algorithm, the number of equivalent classes of cosets of R(1,n) can be computed up to 10 variables. Furthermore, the number of equivalent classes on $\mathcal {R}(s,n)/\mathcal {R}(k,n)$ can also be computed when − 1 ≤ k < s ≤ n ≤ 10, which is a major improvement and advancement comparing to previous methods.

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.

It is proved that if G is an $$\mathfrak {X}$$ -group of infinite rank whose proper subgroups of infinite rank are Baer groups, then so are all proper subgroups of G, where $$\mathfrak {X}$$ is the class defined by N.S. Cernikov as the closure of the class of periodic locally graded groups by the closure operations $$\varvec{\acute{P}}$$ , $$\varvec{\grave{P}}$$ and $$ \varvec{L}$$ . We prove also that if a locally graded group, which is neither Baer nor Cernikov, satisfies the minimal condition on non-Baer subgroups, then it is a Baer-by-Cernikov group which is a direct product of a p-subgroup containing a minimal non-Baer subgroup of infinite rank, by a Cernikov nilpotent $$p^{\prime }$$ -subgroup, for some prime p. Our last result states that a group is locally graded and has only finitely many conjugacy classes of non-Baer subgroups if, and only if, it is Baer-by-finite and has only finitely many non-Baer subgroups.