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The purpose of this note is to revisit Morawetz's method for obtaining a lower bound on the rate of exponential decay of waves for the Dirichlet problem outside star-shaped obstacles, and to discuss the uniqueness of the sphere as the extremiser of the sharp lower bound proved by Ralston. [ABSTRACT FROM AUTHOR]

Let X be a projective variety of dimension n ≥ 1 over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of X. Let G be a group of zero entropy automorphisms of X and G 0 the set of elements in G which are isotopic to the identity. We show that after replacing G by a suitable finite-index subgroup, G / G 0 is a unipotent group of the derived length at most n − 1. This result was first proved by Dinh et al. for compact Kähler manifolds. [ABSTRACT FROM AUTHOR]

Recently, symplectic totally isotropic subspace inclusion graph ℐ n () on a 2 ν -dimensional symplectic space was introduced in [J. He, S. Zhang and G. Zhang, Symplectic totally isotropic subspace inclusion graph, Linear Multilinear Algebra (2022), doi:10.1080/03081087.2022.2112016], which is a graph whose vertices are all totally isotropic subspaces of and two distinct vertices are adjacent if and only if one is contained in the other. In that paper, the authors studied the diameter, girth, clique number and chromatic number of ℐ n () and also studied some other properties of ℐ n (). In this paper, we determine the automorphism group of ℐ n (). [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]

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]

Presenting sporadic simple groups as an automorphism groups of designs and graphs is an exciting field in finite group theory. In this paper, with two different methods, we present some new resolvable simple 3-designs with Higman-Sims sporadic simple group HS as the full automorphism group. Also, we classify all block-transitive self-orthogonal designs on 176 points with even block size that admit sporadic simple group HS as an automorphism group. Furthermore, with these methods we construct some new resolvable 3-designs on 36, 40, 120 and 176 points. [ABSTRACT FROM AUTHOR]

Consider G to be a finite group and p to be a prime divisor of the order | G | in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro's vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ ∈ I r r (G) have a positive height, and let there be an anchor group A ϕ . We prove that if the normalizer N G (A ϕ) is p-constrained, then O p ´ (N G (A ϕ)) ≠ { 1 G } , where O p ´ (N G (A ϕ)) is the maximal normal p ´ subgroup of N G (A ϕ) . We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results. [ABSTRACT FROM AUTHOR]

In this paper, an intrinsic description of some families of linear codes with symmetries is given, showing that they can be described more generally as quasi-group codes, that is, as linear codes allowing a group of permutation automorphisms which acts freely on the set of coordinates. An algebraic description, including the concatenated structure, of such codes is presented. This allows to construct quasi-group codes from codes over rings, and vice versa. The last part of the paper is dedicated to the investigation of self-duality of quasi-group codes. [ABSTRACT FROM AUTHOR]

This paper aims to prove that flag-transitive automorphism groups of 2 - (v , k , λ) designs with λ ≥ (r , λ) 2 and k odd are not of product action type. [ABSTRACT FROM AUTHOR]

In this paper, we first give a relationship between generalized algebraic geometry codes (GAG codes) and algebraic geometry codes (AG codes). More precisely, we show that a GAG code is contained (up to isomorphism) in a suitable AG code. Next we recall the concept of an N1N2-automorphism group, a subgroup of the automorphism group of a GAG code. With the use of the relation we obtained between these two classes of codes, we show that the N1N2-automorphism group is a subgroup of the automorphism group of an AG code. [ABSTRACT FROM AUTHOR]

A Cayley graph Cay (G , S) on a finite group G is said to be normal if the right regular representation R (G) of G is normal in the full automorphism group of Cay (G , S). In this paper, we classify all connected pentavalent non-normal Cayley graphs of order 2 p 2 with solvable automorphism groups and p > 3 a prime. [ABSTRACT FROM AUTHOR]

The groups dVn are an infinite family of groups, first introduced by C. Martínez-Pérez, F. Matucci and B. E. A. Nucinkis, which includes both the Higman-Thompson groups Vn (D 1Vn) and the Brin-Thompson groups nV (D nV2). A description of the groups Aut.Gn;r / (including the groups Gn;1 D Vn) has previously been given by C. Bleak, P. Cameron, Y. Maissel, A. Navas, and F. Olukoya. Their description uses the transducer representations of homeomorphisms of Cantor space introduced in a paper of R. I. Grigorchuk, V.V. Nekrashevich, and V. I. Sushchanskii, together with a theorem of M. Rubin. We generalise the transducers of the latter paper and make use of these transducers to give a description of Aut.dVn/which extends the description of Aut.1Vn/given in the former paper. We make use of this description to show that Out.dV2/Š Out.V2/o Sd, and more generally give a natural embedding of Out.dVn/into Out.Gn;n1 oSd. [ABSTRACT FROM AUTHOR]

In this paper, we determine all simple block-transitive 3-designs on 65 points with block stabilizer of order greater than 2, admitting the 2-transitive action of the group S z (8). As a result, we find 23 3-designs on 65 points that have S z (8) as an automorphism group. Also we construct four 3-(3 6 , 1 8 , λ 1 ) designs for λ 1 ∈ { 7 6 8 , 5 1 8 4 , 6 9 1 2 , 9 2 1 6 } and two 3-(3 6 , 1 7 , λ 2 ) designs for λ 2 ∈ { 2 8 8 0 , 8 6 4 0 }. All of these designs have S 6 (2) as the full automorphism group. Some of these designs are flag and anti-flag-transitive. [ABSTRACT FROM AUTHOR]

In the work by Li (J. Combin. Theory Ser. B , 99 (2009), 447–454.), the author characterized the classification of vertex transitive embeddings of complete graphs, and proposed how to enumerate such maps. In this paper, we study the counting problem of orientable vertex imprimitive complete maps, which is the automorphism group of this map acts imprimitively on its vertex set. Moreover, we obtain the number of non-isomorphic embeddings when the vertex-stabilizer subgroups of the automorphism groups of maps are isomorphic to Z p − 1 with odd prime p. [ABSTRACT FROM AUTHOR]

A family U of non-empty subsets of a set D is called an upfamily if for each set U E U any set F U U belongs to U. The upfamily extension υ(D) of D consists of all upfamilies on D. Any associative binary operation ∗: D×D → D can be extended to an associative binary operation ... In the paper, we show that the upfamily extension (υ(D), -|, -|) of a (strong) doppelsemigroup (D, -|, -|) is a (strong) doppelsemigroup as well and study some properties of this extension. Also we introduce the upfamily functor in the category DSG whose objects are doppelsemigroups and morphisms are doppelsemigroup homomorphisms. We prove that the automorphism group of the upfamily extension of a doppelsemigroup (D, -|, -|) of cardinality |D| ≥ 2 contains a subgroup, isomorphic to C2 × Aut(D, -|, -|). Also we describe the structure of upfamily extensions of all two-element doppelsemigroups and their automorphism groups. [ABSTRACT FROM AUTHOR]

The hull of a linear code plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. Regarding the quantum error correction, linear codes with determined hull are used to construct quantum codes. In this paper, we focus on the hull of Simplex codes and punctured Simplex codes. We firstly study the properties of the matrix produced by the column vectors of a projective space and determine the Euclidean and Hermitian hull of punctured Simplex codes completely. Secondly, we investigate the Euclidean and Hermitian hull of several classes of linear codes from Simplex codes. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the classification of 2-designs admitting a flag-transitive automorphism group. Let be a 2- (v , k , λ) design with λ prime and G be a flag-transitive, point-primitive automorphism group of . We prove that if the socle of G is an alternating group, then has parameters (v , k , λ) = (1 5 , 7 , 3) , (6 , 3 , 2) , (1 0 , 6 , 5) or (1 0 , 4 , 2). [ABSTRACT FROM AUTHOR]

In this paper, we give a classification of extremal Type II Z 4 -codes of length 24. This is equivalent to a classification of 4-frames of the Leech lattice up to the action of the automorphism group. [ABSTRACT FROM AUTHOR]

In this paper we construct distance-regular graphs admitting a vertex transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups M 11 , M 12 , M 22 , M 23 and M 24 . From the binary code spanned by an adjacency matrix of the strongly regular graph with parameters (176,70,18,34) we obtain block designs having the full automorphism groups isomorphic to the Higman-Sims finite simple group. Moreover, from that code we obtain eight 2-designs having the full automorphism group isomorphic to M 22 , whose existence cannot be explained neither by the Assmus-Mattson theorem nor by a transitivity argument. Further, we discuss a possibility of permutation decoding of the codes spanned by adjacency matrices of the graphs constructed and find small PD-sets for some of the codes. [ABSTRACT FROM AUTHOR]

This paper, which is the second of a series of three papers, studies dynamical properties of elements of \operatorname {Out}(F_{\mathtt n}), the outer automorphism group of a nonabelian free group F_{\mathtt n}. We prove that, for every exponentially growing outer automorphism of F_{\mathtt n}, there exists a preferred compact topological space, the space of currents relative to a malnormal subgroup system, on which \phi acts by homeomorphism with a North-South dynamics behavior. [ABSTRACT FROM AUTHOR]

In this paper we settle the question of whether a finite-dimensional vector space V over F p , with p an odd prime, and the family of all the k-sets of elements of V summing up to a given element x, form a 1- (v , k , λ 1) or a 2- (v , k , λ 2) block design, and, in either case, we find a closed form for λ i and characterize the automorphism group. The question is discussed also in the case where the elements of the k-sets are required to be all nonzero, as the two cases happen to be intrinsically inseparable. The "twin case" p = 2 , which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author. [ABSTRACT FROM AUTHOR]

Let \Lambda (f) = K[x][y; f\frac {d}{dx} ] be an Ore extension of a polynomial algebra K[x] over a field K of characteristic zero where f\in K[x]. For a given polynomial f, the automorphism group of the algebra \Lambda (f) is explicitly described. The polynomial case \Lambda (0) = K[x,y] and the case of the Weyl algebra A_1= K[x][y; \frac {d}{dx} ] were done by Jung [J. Reine Angew. Math. 184 (1942), pp. 161–174] and van der Kulk [Nieuw Arch. Wisk. (3) 1 (1953), pp. 33–41], and Dixmier [Bul. Soc. Math. France 96 (1968), pp. 209–242], respectively. Alev and Dumas [Comm. Algebra 25 (1997), pp. 1655–1672] proved that the algebras \Lambda (f) and \Lambda (g) are isomorphic iff g(x) = \lambda f(\alpha x+\beta) for some \lambda, \alpha \in K\backslash \{ 0\} and \beta \in K. Benkart, Lopes and Ondrus [Trans. Amer. Math. Soc. 367 (2015), pp. 1993–2021] gave a complete description of the set of automorphism groups of algebras \Lambda (f). In this paper we complete the picture, i.e. given the polynomial f we have the explicit description of the automorphism group of \Lambda (f). The key concepts in finding the automorphism groups are the eigenform, the eigenroot and the eigengroup of a polynomial (introduced in the paper; they are of independent interest). [ABSTRACT FROM AUTHOR]

Consider the n-dimensional projective space PG (n , q) over a finite field with q elements. A spread in PG (n , q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A deficiency one parallelism is a partial parallelism with one spread less than the parallelism. A transitive deficiency one parallelism corresponds to a parallelism possessing an automorphism group which fixes one spread and is transitive on the remaining spreads. Such parallelisms have been considered in many papers. As a result, an infinite family of transitive deficiency one parallelisms of PG (n , q) has been constructed for odd q, and it has been proved that the deficiency spread of a transitive deficiency one parallelism must be regular, and its automorphism group should contain an elation subgroup of order q 2 . In the present paper we construct parallelisms of PG (3 , 7) invariant under an elation group of order 49 with some additional properties, and thus we succeed to obtain all (46) transitive deficiency one parallelisms of PG (3 , 7) . The three parallelisms from the known infinite family are among them. As a by-product, we also construct a much bigger number (55,022) of parallelisms which have the same spread structure, but are not transitive deficiency one. [ABSTRACT FROM AUTHOR]

In this paper, we study a flag-transitive automorphism group G of symmetric 2- (v , k , λ) designs, where λ > 1 and the order n = k - λ is a prime number. We prove that G is point-primitive, and it is of affine or almost simple type. [ABSTRACT FROM AUTHOR]

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study all tetravalent half-arc-transitive graphs of order 12p, where p is a prime. [ABSTRACT FROM AUTHOR]

In this paper, we explore connected Cayley graphs on non-abelian groups of order 5p², where p is a prime greater than 5, and Sylow p-subgroup is cyclic with respect to tetravalent sets that encompass elements with different orders. We prove that these graphs are normal; however, they are not normal edge-transitive, arc-transitive, nor half-transitive. Additionally, we establish that the group is a 5-CI-group. [ABSTRACT FROM AUTHOR]

We prove that if L = F 4 2 ⁢ (2 2 ⁢ n + 1) ′ and 푥 is a nonidentity automorphism of 퐿, then G = ⟨ L , x ⟩ has four elements conjugate to 푥 that generate 퐺. This result is used to study the following conjecture about the 휋-radical of a finite group. Let 휋 be a proper subset of the set of all primes and let 푟 be the least prime not belonging to 휋. Set m = r if r = 2 or 3 and m = r − 1 if r ⩾ 5 . Supposedly, an element 푥 of a finite group 퐺 is contained in the 휋-radical O π ⁡ (G) if and only if every 푚 conjugates of 푥 generate a 휋-subgroup. Based on the results of this and previous papers, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, unitary simple group, or to one of the groups of type B 2 2 ⁢ (2 2 ⁢ n + 1) , G 2 2 ⁢ (3 2 ⁢ n + 1) , F 4 2 ⁢ (2 2 ⁢ n + 1) ′ , G 2 ⁢ (q) , or D 4 3 ⁢ (q) . [ABSTRACT FROM AUTHOR]

This paper contributes to the classification of flag-transitive symmetric 2- (v , k , λ) designs with λ prime. We investigate the structure of flag-transitive, point-quasiprimitive automorphism groups (G) of such 2-designs by applying the classification of quasiprimitive permutation groups. It is shown that the automorphism groups (G) have either an abelian socle or a non-abelian simple socle. Moreover, according to the classification of finite simple groups, we demonstrate that point-quasiprimitivity implies point-primitivity of G, except when the socle of G is P S L n (q) . [ABSTRACT FROM AUTHOR]

This paper provides a complete determination of which of the alternating groups A n and the symmetric groups S n occur as the automorphism group of some regular or chiral map on an orientable surface, and which of them occur as the automorphism group of a regular map on a non-orientable surface. The situation for some given types (m , k) is also considered, where k is the valency and m is the face-size, with special focus on types with m = 3 , and more particularly with (m , k) = (3 , 7) or (3 , 8) , or their duals. Some observations are made also about what happens for regular and orientably-regular maps with given valency, and for regular and chiral polyhedra. Much but certainly not all of what is presented follows from theorems in previous papers by the author and others, and this one brings them and some new observations together into a single reference. [ABSTRACT FROM AUTHOR]

In this paper, we have found the automorphism group of the Zappa-Szép product of two groups fixing a subgroup. We have computed these automorphisms for a subgroup of order p2 of a group G which is the Zappa-Szép product of two cyclic groups in which one is of order p2 and other is of order m. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the classification of block-transitive 3-designs. Let D = (P , B) be a nontrivial 3- (v , 4 , λ) design and G ≤ A u t (D) acts block-transitively on D with sporadic or alternating socle, then there are exactly 6 incomplete designs as follows: D is isomorphic to a 3- (12 , 4 , λ) design with λ ∈ { 3 , 6 } , and G ≅ M 11 ; D is isomorphic to a 3- (22 , 4 , λ) design with λ ∈ { 3 , 16 } , and S o c (G) = M 22 ; D is isomorphic to a 3-(10, 4, 1) design, and G ≅ M 10 , P G L 2 (9) or P Γ L 2 (9) ; D is isomorphic to a 3-(10, 4, 6) design, and S o c (G) = A 6 . [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension ≤ 2 , hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group Aut (X) of a negatively curved locally finite 2 -dimensional building X is a hyperbolic TDLC-group, whenever Aut (X) acts with finitely many orbits on X. Examples where this result applies include hyperbolic Bourdon's buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension 2 when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups. [ABSTRACT FROM AUTHOR]

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This paper presents a criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves and describes its application via a novel example of a plane curve with non-collinear Galois points. In addition, this paper presents a new characterisation of the Fermat curve in terms of non-collinear Galois points. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the classification of parallelisms in three-dimensional projective spaces over small finite fields of order q by computer. The smallest space in which parallelisms have not yet been classified is for q = 4. Partial results are available. The parallelisms admitting a nontrivial automorphism of odd prime order are known. Moreover, much is known about the case of parallelisms of PG (3 , 4) whose automorphism group is a two group. Namely, everything is known for two of the three possible groups of order two, as well as for cyclic groups of order 4. The present paper will settle the case of parallelisms whose automorphism group is elementary abelian of order 4. This leaves open the cases of parallelisms whose full automorphism groups are either trivial or a specific group of order two. [ABSTRACT FROM AUTHOR]

Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed storage as well as theoretical interest. However, not much work on asymptotical behavior of locally repairable codes has been done until now. In particular, there is little result on constructive lower bound of asymptotical behavior of locally repairable codes. In this paper, we extend the construction given by Barg et al. via automorphism groups of function field towers. The main advantage of our construction is to allow more flexibility of locality. Furthermore, we show that the Gilbert–Varshamov type bound on locally repairable codes can be improved for all sufficiently large alphabet size $q$. [ABSTRACT FROM AUTHOR]

This review paper is a continuation of two previous review papers devoted to properties of modules over discrete valuation domains. The first part of this work was published in the volumes 121 (Part I, Chaps. 1–4, Secs. 1–22) and 122 (Part II, Chaps. 5–8, Secs. 23–39) of the series Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory (English translation: Journal of Mathematical Sciences (New York), vol. 145, No. 4, pp. 4997–5117 (2007), and vol. 151, No. 5, pp. 3225–3371 (2008)). In this review paper, we preserve the numeration of chapters and sections of parts I and II. The first part consists of Chapter 9 "Appendix," Secs. 40–42. In Sec. 40, we consider p-adic torsion-free modules with isomorphic automorphism groups. Section 41 is devoted to torsion-free modules over a complete discrete valuation domain with isomorphic radicals of their endomorphism rings. The volume of the paper does not allow us to provide the proofs of all the results that appeared after publication of the previous parts and directly related to the issues under consideration in it. In the final Sec. 42, we describe some of these new results. [ABSTRACT FROM AUTHOR]

In a 2021 paper, Conder et al. constructed two infinite families of chiral 4-polytopes of type { 4 , 4 , 4 } with solvable automorphism groups. Here we present a general construction for chiral polytopes of type { 4 , 4 , ... , 4 } with rank 4, 5 and 6, which are obtained as Boolean covers of the unique tight regular polytope of the same type. [ABSTRACT FROM AUTHOR]

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. Let N be a normal subgroup of G. For x ∈ N, we have xG ⊆ N and xG is called a G-class of the normal subgroup N. In this paper, we develop several results on the p -part of G-class sizes of a normal subgroup of a finite group G. [ABSTRACT FROM AUTHOR]

Let IAn denote the group of IA-automorphisms of a free group of rank n, and let Ibn denote the Torelli subgroup of the mapping class group of an orientable surface of genus n with b boundary components, b D 0; 1. In 1935, Magnus proved that IAn is finitely generated for all n, and in 1983, Johnson proved that Ibn is finitely generated for n ≥ 3. It was recently shown that for each k ∈ N, the k-th terms of the lower central series ykIAn and ykIbn are finitely generated when n ≫-k; however, no information about finite generating sets was known for k > 1. The main goal of this paper is to construct an explicit finite generating set for Y2IAn = [IAn, IAn] and almost explicit finite generating sets for Y2Ibn and the Johnson kernel, which contains Y2Ibn as a finite index subgroup. [ABSTRACT FROM AUTHOR]

A graph is edge-primitive if its automorphism group acts primitively on the edge set of the graph. Edge-primitive graphs form an important subclass of symmetric graphs. In this paper, edge-primitive graphs of order as a product of two distinct primes are completely determined. This depends on non-abelian simple groups with a subgroup of index pq being classified, where p > q are odd primes. [ABSTRACT FROM AUTHOR]

Consider G to be a simple graph with n vertices and m edges, and L (G) to be a Laplacian matrix with Laplacian eigenvalues of μ 1 , μ 2 , ... , μ n = z e r o . Write S k (G) = ∑ i = 1 k μ i as the sum of the k-largest Laplacian eigenvalues of G, where k ∈ { 1 , 2 , ... , n } . The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer's conjecture states that S k (G) ≤ m + k + 1 2 , where k = 1 , 2 , ... , n . This paper proves Brouwer's conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph W n + 1 , which provides a better approximation for this upper bound using Brouwer's conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of W n + 1 , n ≥ 3 . We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves. [ABSTRACT FROM AUTHOR]

The present paper investigates 16-dimensional compact translation planes with automorphism groups of dimension d between 35 and 37; planes with groups of higher dimensions have been classified by Hähl. We obtain a complete classification for d = 37 (up to isomorphisms). It turns out that these planes have Lenz type V and are already described in a recent paper of Hähl and Meyer [10]. Moreover, we give a partial classification for d = 35 and d = 36. The latter case will be completely finished in a forthcoming paper [16] of the author, while the case where d = 35 is completed except for groups whose maximal compact subgroups are 9-dimensional. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum E n , whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that E n is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group G n of the formal group in question. In this paper we find that the G n -equivariant dual of E n is in fact E n twisted by a sphere with a non-trivial (when n > 1 ) action by G n . This sphere is a dualizing module for the group G n , and we construct and study such an object I G for any compact p-adic analytic group G . If we restrict the action of G on I G to certain type of small subgroups, we identify I G with a specific representation sphere coming from the Lie algebra of G . This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of E n hH for select choices of p and n and finite subgroups H of G n . [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]

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]

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]