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We study 2- ( v , k , λ ) (v,k,\lambda) designs D = ( P , B ) \mathcal{D}=(\mathcal{P},\mathcal{B}) admitting a flag-transitive automorphism group 퐺, where k = p 2 k=p^{2} or p 3 p^{3} , and 푝 is prime. We prove that if k = p 2 k=p^{2} , then 퐺 must be point-primitive, and 퐺 is of affine or almost simple type except for two examples. If k = p 3 k=p^{3} , and 퐺 is point-primitive, then 퐺 is of affine, almost simple, or product type with G ≤ H ≀ S 2 G\leq H\wr S_{2} acting on P = Δ × Δ \mathcal{P}=\Delta\times\Delta , where 퐻 is a primitive group on Δ. Furthermore, if 퐺 is point-imprimitive with a system Σ of imprimitivity consisting of 푑 classes of size 푐, we determine the parameters ( v , k , μ , c , d ) (v,k,\mu,c,d) , where 휇 is the size of the nonempty set B ∩ C B\cap C with B ∈ B B\in\mathcal{B} and C ∈ Σ C\in\Sigma . Moreover, some infinite families of flag-transitive 2- ( v , p 3 , λ ) (v,p^{3},\lambda) designs are also constructed. [ABSTRACT FROM AUTHOR]

This article, published in the Journal of Mathematical Sciences, explores the computation of the component group of an arbitrary real algebraic group. The author demonstrates that the component group is always an elementary Abelian 2-group. The computation is based on structure results on algebraic groups and Galois cohomology methods. The paper provides necessary material on algebraic groups and Galois cohomology, and includes an example of the real locus of an elliptic curve. This document discusses the computation of the group of connected components of the real locus of a connected algebraic group defined over R. It introduces the concepts of cocycles, exact sequences of groups, and Galois cohomology. The main result of the document is a formula for computing the group of connected components, which involves lattices and coroot lattices. The proof of the formula is similar to a previous proof in another paper. The given text discusses the component groups of certain mathematical structures, specifically linear algebraic groups and Abelian varieties. It presents theorems and examples related to these component groups, providing mathematical proofs and explanations. The text explores different cases and scenarios, such as when the period lattice is preserved by complex multiplication or when it is multiplied by certain values. The examples given include elliptic curves and their connected components. The text is written in a technical and mathematical language, and it may be useful for researchers studying these specific topics. [Extracted from the article]

The article explores the conditions under which a specific type of variety, known as a trinomial variety, will have a finite number of automorphism group orbits. Trinomial varieties are defined as the spectrum of trinomial algebras, which are based on certain data. The authors provide examples of classes of varieties that have a finite number of orbits, such as algebraic groups, toric varieties, and spherical varieties. They also examine trinomial hypersurfaces and generalize previous results to prove a sufficient condition for the finiteness of the number of orbits on a trinomial variety. The authors discuss the connection between locally nilpotent derivations and automorphisms of the variety, and provide preliminary information on locally nilpotent derivations and trinomial varieties. The text is technical in nature and may be of interest to researchers studying algebraic geometry. [Extracted from the article]

The article titled "Gradings of Galois Extensions" explores the concept of gradings in finite field extensions, specifically focusing on fine gradings. The authors investigate fine gradings of Kummer extensions and describe all possible gradings of Kummer extensions. They also examine a wider class of Galois extensions that admit fine gradings and discuss their applications in computing Galois groups. The article provides definitions, propositions, and lemmas to support their findings. The text discusses various cases and propositions related to Kummer extensions and Galois extensions, providing proofs and explanations for each proposition. It also discusses nonstandard gradings of Kummer extensions and provides criteria for their existence. The text concludes by discussing the coarsening of a fine grading of a Galois extension. Overall, the text provides a detailed analysis of fine gradings in field extensions and their implications. [Extracted from the article]

A graph is edge‐transitive if its automorphism group acts transitively on the edge set. This paper presents a complete classification for connected edge‐transitive cubic graphs of order 2 n $2n$, where n $n$ is even and square‐free. In particular, it is shown that such a graph is either symmetric or isomorphic to one of the following graphs: a semisymmetric graph of order 420, a semisymmetric graph of order 29,260, and five families of semisymmetric graphs constructed from the simple group PSL 2( p ) ${{\bf{\text{PSL}}}}_{2}(p)$. [ABSTRACT FROM AUTHOR]

The Kuratowski graphs K 3 , 3 and K 5 characterize planarity. Counting distinct 2-cell embeddings of these two graphs on orientable surfaces was previously done by Mull (1999) and Mull et al. (2008), using Burnside's Lemma and automorphism groups of K 3 , 3 and K 5 , without actually constructing the embeddings. We obtain all 2-cell embeddings of these graphs on the double torus, using a constructive approach. This shows that there is a unique non-orientable 2-cell embedding of K 3 , 3 , and 14 orientable and 17 non-orientable 2-cell embeddings of K 5 on the double torus, which are explicitly obtained using an algorithmic procedure of expanding from minors. Therefore we confirm the numbers of embeddings obtained by Mull (1999) and Mull et al. (2008). As a consequence, several new polygonal representations of the double torus are presented. Rotation systems for the one-face embeddings of K 5 on the triple torus are also found, using exhaustive search. [ABSTRACT FROM AUTHOR]

In this paper, we introduce several new construction techniques of linear complimentary dual (LCD) codes. First, we show that if a LCD code is fixed by a transitive automorphism group, then the punctured code is also LCD. We show that several important families such as anti-primitive BCH codes and extended Gabidulin codes are LCD. Finally relationships among self-dual codes, λ-orthogonal matrices and LCD codes are studied. As an application we give LCD codes from Paley-type bipartite graph and from punctured MacDonalds codes. Further, we give a bound on some LCD codes. [ABSTRACT FROM AUTHOR]

The symmetric 2- (v , k , λ) designs with k > λ (λ − 3) / 2 admitting a flag-transitive point-imprimitive automorphism group are completely classified: they are the known 2-designs with parameters (16 , 6 , 2) , (45 , 12 , 3) , (15 , 8 , 4) or (96 , 20 , 4). [ABSTRACT FROM AUTHOR]

For any almost-simple group G over an algebraically closed field k of characteristic zero, we describe the automorphism group of the moduli space of semistable G -bundles over a connected smooth projective curve C of genus at least 4. The result is achieved by studying the singular fibers of the Hitchin fibration. As a byproduct, we provide a description of the irreducible components of two natural closed subsets in the Hitchin basis: the divisor of singular cameral curves and the divisor of singular Hitchin fibers. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the classification of all pairs (T,G) $({\mathscr{T}},G)$, where T ${\mathscr{T}}$ is a triple system and G $G$ is a block‐transitive but not flag‐transitive automorphism group of T ${\mathscr{T}}$. We prove that if the socle of G $G$ is a sporadic or alternating group, then one of the following holds: (i)T ${\mathscr{T}}$ is a TS(10,2) $TS(10,2)$ and G≅A5 $G\cong {A}_{5}$;(ii)T ${\mathscr{T}}$ is a TS(10,4) $TS(10,4)$ and G≅S5 $G\cong {S}_{5}$;(iii)T ${\mathscr{T}}$ is a TS(55,28) $TS(55,28)$ and G≅A11 $G\cong {A}_{11}$ or S11 ${S}_{11}$;(iv)T ${\mathscr{T}}$ is a TS(55,λ) $TS(55,\lambda)$ with λ∈{4,8,16} $\lambda \in \{4,8,16\}$ and G≅M11 $G\cong {M}_{11}$. [ABSTRACT FROM AUTHOR]

Given a construction f on groups, we say that a group G is f -realisable if there is a group H such that G ≅ f (H) , and completely f -realisable if there is a group H such that G ≅ f (H) and every subgroup of G is isomorphic to f (H 1) for some subgroup H 1 of H and vice versa. In this paper, we determine completely Aut -realisable groups. We also study f -realisable groups for f = Z , F , M , D , Φ , where Z (H) , F (H) , M (H) , D (H) and Φ (H) denote the center, the Fitting subgroup, the Chermak–Delgado subgroup, the derived subgroup and the Frattini subgroup of the group H , respectively. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we use the Balinsky-Novikov construction to naturally realize the super Heisenberg-Virasoro algebra g . We then proceed to compute the second cohomology of g with trivial coefficients and determine its universal central extension. This yields two odd 2-cocycles in addition to the Virasoro 2-cocycle. We also calculate the first cohomology group of g in the adjoint module to classify all derivations of g . Finally, we classify all automorphisms of g and describe its automorphism group. [ABSTRACT FROM AUTHOR]

This paper is a continuation of Ref. 7. In this paper, we give an explicit construction of a non-Bichon type co-action on a multigraph that is, it preserves quantum symmetry of (V,E) in our sense but not always in Bichon’s sense.2 This construction itself is motivated from automorphisms of quantum graphs. [ABSTRACT FROM AUTHOR]

In this paper, we first show that the monoid of separable surjective self-morphisms of a variety of Ueno type coincides with the group of automorphisms. We also give an explicit description of the automorphism group. As applications, we confirm Kawaguchi–Silverman conjecture for automorphisms of a variety of Ueno type and some Calabi–Yau three-fold, defined over ℚ¯. [ABSTRACT FROM AUTHOR]

Assume that H is a finite group that has a normal 2 -complement. Under some conditions, it is proven that the normalizer property holds for H. In particular, if there is a nilpotent subgroup of index 2 in H , then H has the normalizer property. The result of Li, Sehgal and Parmenter, stating that the normalizer property holds for finite groups that have an abelian subgroup of index 2 is generalized. [ABSTRACT FROM AUTHOR]

AbstractGiven positive integers p and m, where p is assumed to be an odd prime, we determine the automorphism groups of p-groups J, H, and K of orders p7m, p6m, and p5m, and nilpotency classes 5, 4, and 3, respectively, all of which arise naturally from the Macdonald group 〈x,y|x[x,y]=x1+pmℓ,y[y,x]=y1+pmℓ〉, where ℓ∈Z is relatively prime to p. [ABSTRACT FROM AUTHOR]

In this work, we present the analytical closed forms of the Balaban index for anthracene and catacondensed benzenoid systems using group theoretic techniques. The Balaban index is a distance-based topological index that provides valuable information about the properties of chemical structures. We emphasize the importance of determining analytical closed forms of the Balaban index for catacondensed benzenoid systems and linear chains of anthracene, as it enables a deeper understanding of these systems and their behavior. Our analysis utilizes the group action of the automorphism group of these chains on the set of vertices, which refer to the points where the chains intersect. In future work, we plan to determine the Balaban index of other polymeric linear chains using group theoretic techniques and extend the applications of this index to other fields, such as materials science and biology. It is clear that the Balaban index remains a valuable tool in theoretical and computational chemistry, and its applications are constantly evolving. [ABSTRACT FROM AUTHOR]

We consider the Macdonald group 〈 x , y | x [ x , y ] = x 1 + 2 m ℓ , y [ y , x ] = y 1 + 2 m ℓ 〉 and its Sylow 2-subgroup J = 〈 x , y | x [ x , y ] = x 1 + 2 m ℓ , y [ y , x ] = y 1 + 2 m ℓ , x 2 3 m − 1 = y 2 3 m − 1 = 1 〉 , where m ≥ 1 and ℓ is odd. Then J has order 2 7 m − 3 , and nilpotency class 5 if m > 1 and 3 if m = 1. We determine the automorphism group of the 2-groups J, H = J / Z (J) and K = H / Z (H) , where | H | = 2 6 m − 3 and | K | = 2 5 m − 3 . Explicit multiplication, power, and commutator formulas for J, H and K are given, and used in the calculation of Aut (J) , Aut (H) and Aut (K). [ABSTRACT FROM AUTHOR]

Using a refinement of the differential method introduced by Oguiso and Yu, we provide effective conditions under which the automorphisms of a smooth degree d hypersurface of \mathbf {P}^{n+1} are given by generalized triangular matrices. Applying this criterion we compute all the remaining automorphism groups of Klein hypersurfaces of dimension n\geq 1 and degree d\geq 3 with (n,d)\neq (2,4). We introduce the concept of extremal polarized Hodge structures, which are structures that admit an automorphism of large prime order. Using this notion, we compute the automorphism group of the polarized Hodge structure of certain Klein hypersurfaces that we call of Wagstaff type, which are characterized by the existence of an automorphism of large prime order. For cubic hypersurfaces and some other values of (n,d), we show that both groups coincide (up to involution) as predicted by the Torelli Principle. [ABSTRACT FROM AUTHOR]

This paper is devoted to ring theoretic properties and representations of the quantum Galilei group F q (G) and its Hopf dual U q (g) . We classify the primitive ideals and simple modules of F q (G) and U q (g) over an algebraically closed field of arbitrary characteristic. We determine their groups of automorphisms over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

In this short note, we prove the existence of infinitely many pairwise nonisomorphic, non‐hyperelliptic Riemann surfaces with automorphism group acting transitively on the Weierstrass points. We also find all compact Riemann surfaces with automorphism group acting transitively on the Weierstrass points, under the assumption that they are simple. [ABSTRACT FROM AUTHOR]

AbstractWe investigate the action of Schur algebra on the Lie algebras of derivations of free Lie algebras and operad structures constructed from it. We also show that the Lie algebra of derivations is generated by quadratic derivations together with the action of the Schur operad. Applications to certain subgroups of the automorphism group of a finitely generated free group are given as well. [ABSTRACT FROM AUTHOR]

We prove that the automorphism group of a general complete intersection X in ℂ ℙ n is trivial with a few well-understood exceptions. We also prove that the automorphism group of a complete intersection X acts on the cohomology of X faithfully with a few well-understood exceptions. [ABSTRACT FROM AUTHOR]

The generalized quantum Euclidean group O q (b m , n) is a natural generalization of the quantum Euclidean group O q (b 1 , 1) . The Heisenberg double D q (b m , n) of O q (b m , n) is the smash product of O q (b m , n) with its Hopf dual U q (b m , n) . In this paper, we study the weight modules, the prime spectrum and the automorphism group of the Heisenberg double D q (b m , n) . [ABSTRACT FROM AUTHOR]

Let H be the Radford Hopf algebra of dimension twelve over a field k . In this paper, we investigate the automorphism groups of the Green ring r(H) and the Green algebra ℂ(H) ≔ ℂ ⊗ℤr(H) over the complex number field ℂ. The ring automorphisms of r(H) and the algebra automorphisms of ℂ(H) are all described. It is shown that the ring automorphism group of r(H) is isomorphic to the Klein group K4, and the algebra automorphism group of ℂ(H) is isomorphic to the direct product ℂx × S4, where ℂx is the multiplicative group of all nonzero complex numbers and S4 is the symmetric group of degree 4. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we study certain combinatorial attributes of Ind-schemes of polynomial automorphisms in positive characteristic. In particular, we prove that over an algebraically closed field K of positive characteristic ≠2 every automorphism of the group of origin-preserving automorphisms of the polynomial algebra K[x1,…,xn] (n≥3), which fixes every diagonal matrix, preserves, up to composition with a linear inner automorphism, every tame automorphism. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a connected graph with the vertex-set V and the edgeset E. The subdivision graph S(G) of the graph G is obtained from G by adding a vertex in the middle of every edge of G. In this paper, we investigate some properties of the graphs S (G) and L(S (G)), where L(S (G)) is the line graph of S (G). We will see that S (G) and L(S (G)) inherit some properties of G. For instance, we show that if G Cn, then Aut(G) = Aut(L(S(G))) (as abstract groups), where Cn is the cycle of order n. [ABSTRACT FROM AUTHOR]

Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research. [ABSTRACT FROM AUTHOR]

In [V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math. 226(3) (1999) 20–28, Sec. 5, p. 32], Arnold writes: 'Classification of singularities of curves can be interpreted in dual terms as a description of "co-artin" subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)'. In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let K be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras of the polynomial algebra K [ x ] that contains the ideal x m K [ x ] for some m ≥ 1. It is proven that the set = ∐ m , Γ (m , Γ) is a disjoint union of affine algebraic varieties (where Γ ∐ { 0 , m , m + 1 , ... } is the semigroup of the singularity and m − 1 is the Frobenius number). It is proven that each set (m , Γ) is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on (m , Γ). An isomorphism criterion is given for the algebras in . For each algebra A ∈ (m , Γ) , explicit sets of generators and defining relations are given and the automorphism group Aut K (A) is explicitly described. The automorphism group of the algebra A is finite if and only if the algebra A is not isomorphic to a monomial algebra, and in this case | Aut K (A) | < dim K (A / A) where A is the conductor of A. The set of orders of the automorphism groups of the algebras in (m , Γ) is explicitly described. [ABSTRACT FROM AUTHOR]

We give an isomorphism test that runs in time npolylog(h) on all n-vertex graphs excluding some h-vertex graph as a topological subgraph. Previous results state that isomorphism for such graphs can be tested in time npolylog(h) (Babai, STOC 2016) and n{f(h) for some function f (Grohe and Marx, SIAM J. Comp., 2015). Our result also unifies and extends previous isomorphism tests for graphs of maximum degree d running in time npolylog(d) (SIAM J. Comp., 2023) and for graphs of Hadwiger number h running in time npolylog(h) (SIAM J. Comp., 2023). [ABSTRACT FROM AUTHOR]

We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras. [ABSTRACT FROM AUTHOR]

Let φ: G → G be an automorphism of an infinite group G. One has an equivalence relation ~φ on G defined as x ~φ y if there exists a z ∈ G such that y = zxφ(z-1). The equivalence classes are called φ-twisted conjugacy classes, and the set G/~φ of equivalence classes is denoted by R(φ). The cardinality R(φ) of R(φ) is called the Reidemeister number of φ. We write R(φ) = ∞ when R(φ) is infinite. We say that G has the R∞-property if R(φ) = ∞ for every automorphism φ of G. We show that the groups G = GLn(R), SLn(R) have the R∞-property for all n ≥ 3 when F[t] ⊂ R ... F(t), where F is a subfield of Fp. When n ≥ 4, we show that any subgroup H ∈ GLn(R) that contains SLn(R) also has the R∞-property. [ABSTRACT FROM AUTHOR]

Let C ⊂ 핡³ be a canonical curve of genus 4 over an algebraically closed field k of characteristic zero. For a line l, we consider the projection πl C → 핡¹ with center l and the extension of the function fields πl* W (k (핡¹) → k(C). A line l is referred to as a cyclic line if the extension πl* W (k ((핡¹)) is cyclic. A line l ⊂ 핡³ is said to be skew if C ∩ l = θ. We prove that the number of skew cyclic lines is equal to 0; 1; 3 or 9. We determine curves that have nine skew cyclic lines. [ABSTRACT FROM AUTHOR]

We show that there is a distortion element in a finitely generated subgroup G of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of G , and that a sofic shift's automorphism group contains a distortion element if and only if the sofic shift is uncountable. We obtain also that groups of Turing machines and the higher-dimensional Brin–Thompson groups $mV$ admit distortion elements; in particular, $2V$ (unlike V) does not admit a proper action on a CAT $(0)$ cube complex. In each case, the distortion element roughly corresponds to the SMART machine of Cassaigne, Ollinger, and Torres-Avilés [A small minimal aperiodic reversible Turing machine. J. Comput. System Sci. 84 (2017), 288–301]. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of the lattice of c-structures. We define the concept of simple expansion and examine some of its properties. Additionally, we determine the automorphism group of the lattice of c-structures and explore the fixed points of the group of automorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group G acting on another group K equipped with a filtration indexed by a "good" ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid ℕ 2 of pairs on non-negative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group ℳ of a surface Σ g , 1 with one boundary component, equipped with the normal subgroups X ̄ , Ȳ of π 1 (Σ g , 1) associated to a standard Heegaard splitting of the 3 -sphere. We also consider the case where the group G is the automorphism group of a free group. [ABSTRACT FROM AUTHOR]

We study the order of automorphism groups of cyclic covering fibrations of a ruled surface. Arakawa and later Chen studied it for hyperelliptic fibrations and gave the upper bound. The purpose of this paper is to pursue the analog for cyclic covering fibrations. [ABSTRACT FROM AUTHOR]

In a previous paper the authors with M. Conder proved that the alternating group A n and the symmetric group S n are pseudo-Hurwitz for any n > 23 , as well as are A 18 , A 19 , A 21 , A 22 and A 23 , and also S 10 , S 17 , S 19 , S 20 , S 21 and S 22. In this paper, we provide geometric tools for constructing all pseudo-Hurwitz groups up to a "small" degree. This construction, applied up to degree 23, shows that there are no other degrees for which A n and S n are pseudo-Hurwitz. In the second part of this paper we show that there are both reflexible and chiral pseudo-Hurwitz maps with automorphism group A n and S n , for every n ⩾ 36. [ABSTRACT FROM AUTHOR]

This work is to understand structures of even lattice vertex operator algebras by using the geometry of the varieties of their semi-conformal vectors. In this paper, we shall describe the varieties of semi-conformal vectors of a class of even lattice vertex operator algebras with rank two to characterize these vertex operator algebras. Moreover, we shall give the G-orbit decomposition of the fixed vertex operator subalgebra V L + of a rank-two lattice vertex operator algebra V L , where G is the automorphism group preserving the conformal vector of V L +. Based on this decomposition, we classify all maximal semi-conformal subalgebras of V L +. [ABSTRACT FROM AUTHOR]

An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group. In this paper we obtain a structure theorem for locally finite quasi-transitive graphs avoiding a minor, which is reminiscent of the Robertson-Seymour Graph Minor Structure Theorem. We prove that every locally finite quasi-transitive graph G avoiding a minor has a tree-decomposition whose torsos are finite or planar; moreover the tree-decomposition is canonical, i.e. invariant under the action of the automorphism group of G. As applications of this result, we prove the following. • Every locally finite quasi-transitive graph attains its Hadwiger number, that is, if such a graph contains arbitrarily large clique minors, then it contains an infinite clique minor. This extends a result of Thomassen (1992) [38] who proved it in the (quasi-)4-connected case and suggested that this assumption could be omitted. In particular, this shows that a Cayley graph excludes a finite minor if and only if it avoids the countable clique as a minor. • Locally finite quasi-transitive graphs avoiding a minor are accessible (in the sense of Thomassen and Woess), which extends known results on planar graphs to any proper minor-closed family. • Minor-excluded finitely generated groups are accessible (in the group-theoretic sense) and finitely presented, which extends classical results on planar groups. • The domino problem is decidable in a minor-excluded finitely generated group if and only if the group is virtually free, which proves the minor-excluded case of a conjecture of Ballier and Stein (2018) [7]. [ABSTRACT FROM AUTHOR]

Given a finite simple graph Γ on n vertices its complementary prism is the graph Γ Γ ¯ that is obtained from Γ and its complement Γ ¯ by adding a perfect matching where each its edge connects two copies of the same vertex in Γ and Γ ¯. It generalizes the Petersen graph, which is obtained if Γ is the pentagon. The automorphism group of Γ Γ ¯ is described for an arbitrary graph Γ. In particular, it is shown that the ratio between the cardinalities of the automorphism groups of Γ Γ ¯ and Γ can attain only the values 1, 2, 4, and 12. It is shown that Γ Γ ¯ is vertex-transitive if and only if Γ is vertex-transitive and self-complementary. Moreover, the complementary prism is not a Cayley graph whenever n > 1. [ABSTRACT FROM AUTHOR]

We prove that every automorphism of the category of free finitely generated dimonoids is quasi-inner. Also we show the automorphism group of the endomorphism semigroup of the free dimonoid on a nonsingleton set is isomorphic to the direct product of the Klein four-group and the symmetric group. [ABSTRACT FROM AUTHOR]

Given a field K , we investigate which subgroups of the group Aut A K 2 of polynomial automorphisms of the plane are linear or not. The results are contrasted. The group Aut A K 2 itself is nonlinear, except if K is finite, but it contains some large subgroups, of "codimension-five" or more, which are linear. This phenomenon is specific to dimension two: it is easy to prove that any natural "finite-codimensional" subgroup of Aut A K 3 is nonlinear, even for a finite field K. When ch K = 0 , we also look at a similar questions for f.g. subgroups, and the results are again disparate. The group Aut A K 2 has a one-related f.g. subgroup which is not linear. However, there is a large subgroup, of "codimension-three", which is locally linear but not linear. [ABSTRACT FROM AUTHOR]

We study structurable algebras and their associated Freudenthal triple systems over commutative rings. The automorphism groups of these triple systems are exceptional groups of type E 7 , and we realize groups of type E 6 as centralizers. When 6 is invertible, we further give a geometric description of homogeneous spaces of type E 7 / E 6 , and show that they parametrize principal isotopes of Brown algebras. As opposed to the situation over fields, we show that such isotopes may be non-isomorphic. [ABSTRACT FROM AUTHOR]

For N≥3, the abstract commensurators of both Aut(FN) and its Torelli subgroup IAN are isomorphic to Aut(FN) itself. [ABSTRACT FROM AUTHOR]

The semidirect discrete logarithm problem (SDLP) is the following analogue of the standard discrete logarithm problem in the semidirect product semigroup G ⋊ End (G) for a finite semigroup G. Given g ∈ G , σ ∈ End (G) , and h = ∏ i = 0 t - 1 σ i (g) for some integer t, the SDLP (G , σ) , for g and h, asks to determine t. As Shor's algorithm crucially depends on commutativity, it is believed not to be applicable to the SDLP. For generic semigroups, the best known algorithm for the SDLP is based on Kuperberg's subexponential time quantum algorithm. Still, the problem plays a central role in the security of certain proposed cryptosystems in the family of semidirect product key exchange. This includes a recently proposed signature protocol called SPDH-Sign. In this paper, we show that the SDLP is even easier in some important special cases. Specifically, for a finite group G, we describe quantum algorithms for the SDLP in G ⋊ Aut (G) for the following two classes of instances: the first one is when G is solvable and the second is when G is a matrix group and a power of σ with a polynomially small exponent is an inner automorphism of G. We further extend the results to groups composed of factors from these classes. A consequence is that SPDH-Sign and similar cryptosystems whose security assumption is based on the presumed hardness of the SDLP in the cases described above are insecure against quantum attacks. The quantum ingredients we rely on are not new: these are Shor's factoring and discrete logarithm algorithms and well-known generalizations. [ABSTRACT FROM AUTHOR]

The Reed–Muller codes are a family of error-correcting codes that have been widely studied in coding theory. In 2020, Yan and Lin introduced a variant of Reed–Muller codes called symmetric Reed–Muller codes. We investigate linear maps of the automorphism group of symmetric Reed–Muller codes and show that the set of these linear maps forms a subgroup of the general linear group, which is the automorphism group of punctured Reed–Muller codes.We provide a method to determine all the automorphisms in this subgroup explicitly for some special cases. [ABSTRACT FROM AUTHOR]

A finite group G $G$ admits an oriented regular representation if there exists a Cayley digraph of G $G$ such that it has no digons and its automorphism group is isomorphic to G $G$. Let m $m$ be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented m $m$‐semiregular representations using m $m$‐Cayley digraphs. Given a finite group G $G$, an m $m$‐Cayley digraph of G $G$ is a digraph that has a group of automorphisms isomorphic to G $G$ acting semiregularly on the vertex set with m $m$ orbits. We say that a finite group G $G$ admits an oriented m $m$‐semiregular representation (Om $m$SR for short) if there exists an m $m$‐Cayley digraph Γ ${\rm{\Gamma }}$ of G $G$ such that it has no digons and G $G$ is isomorphic to its automorphism group. Moreover, if Γ ${\rm{\Gamma }}$ is regular, that is, each vertex has the same in‐ and out‐valency, we say Γ ${\rm{\Gamma }}$ is a regular oriented m $m$‐semiregular representation (regular Om $m$SR for short) of G $G$. In this paper, we classify finite groups admitting a regular Om $m$SR or an Om $m$SR for each positive integer m $m$. [ABSTRACT FROM AUTHOR]