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A gyrogroup is defined as having a binary operation ⋆ containing an identity element such that each element has an inverse. Furthermore, for each pair (a, b) of elements of this structure, there exists an automorphism gyr[a, b] with the property that left associativity and the left loop property are satisfied. Since each gyrogroup is a left Bol loop, some results of Burn imply that all gyrogroups of orders p, 2p, and p², where p is a prime number, are groups. This paper aims to classify gyrogroups of orders 8, 12, 15, 18, 20, 21, and 28. [ABSTRACT FROM AUTHOR]

Assume that A and G are finite groups of coprime orders such that A acts on G via automorphisms. Let p be a prime. The following coprime action version of a well-known theorem of Itô about the structure of a minimal non- p -nilpotent groups is proved: if every maximal A -invariant subgroup of G is p -nilpotent, then G is p -soluble. If, moreover, G is not p -nilpotent, then G must be soluble. Some earlier results about coprime action are consequences of this theorem. [ABSTRACT FROM AUTHOR]

Let S be a prime ring with automorphisms α,β. A bi-additive map D is called an (α,β) Jordan bi-derivation if D(k2,s)=D(k,s)α(k)+β(k)D(k,s). In this paper, we find conditions under which a symmetric (α,β) Jordan bi-derivation becomes a symmetric (α,β) bi-derivation. We also characterize the symmetric (α,β) Jordan bi-derivations. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring and α be an automorphism of R. A mapping F : R → R (not necessarily additive) is called multiplicative generalized (α, α)- derivation if there exists a unique (α, α)-derivation d of R such that F(xy) = F(x)α(y) + α(x)d(y) for all x, y = R. In the present paper, we intend to study several algebraic identities involving multiplicative generalized (α, α)-derivations on appropriate subsets of semiprime rings and collect the information about the commutative structure of these rings. [ABSTRACT FROM AUTHOR]

In this paper, we deduce some hyperstability results for a generalized class of homogeneous Pexiderized functional equations, expressed as ∑ ρ ∈ Γ f x ρ. y = ℓ f (x) + ℓ g (y) , x , y ∈ M , which is inspired by the concept of Ulam stability. Indeed, we prove that function f that approximately satisfies an equation can, under certain conditions, be considered an exact solution. Domain M is a monoid (semigroup with a neutral element), Γ is a finite subgroup of the automorphisms group of M, ℓ is the cardinality of Γ , and f , g : M → G such that (G , +) denotes an ℓ-cancellative commutative group. We also examine the hyperstability of the given equation in its inhomogeneous version ∑ ρ ∈ Γ f x ρ. y = ℓ f (x) + ℓ g (y) + ψ (x , y) , x , y ∈ M , where ψ : M × M → G . Additionally, we apply the main results to elucidate the hyperstability of various functional equations with involutions. [ABSTRACT FROM AUTHOR]

Let R be a 2−torsion free semiprime ring, α an automorphism and U a noncentral square-closed Lie ideal of R. An additive mapping T : R → R is called a left (resp. right) α−centralizer of R if T(xy) = T(x)α(y) (resp. T(xy) = α(x)T(y)) holds for all x, y ∈ R. In this paper, we proved the following result: i) If T (uvu) = α (u) T (v) α (u) for all u, v ∈ U, then T is a left α−centralizer, ii) If T(uvu) = T(u)α(vu) (resp. T(uvu) = α(uv)T(u)) for all u, v ∈ U, then T is a left (resp. right) α-centralizer on U. [ABSTRACT FROM AUTHOR]

A long-standing conjecture asserts that every finite nonabelian p-group has a non-inner automorphism of order p. This paper proves the conjecture for finite p-groups of coclass 4 and 5 (p ≥ 5). We also prove the conjecture for an odd order nonabelian p-group G with cyclic center satisfying CG(Gpγ3(G)) ∩ Z3(G) ≤ Z(Φ(G)). [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 study non-symplectic automorphisms of odd order on K3 surfaces which are not purely. In particular we shall describe fixed loci of non-symplectic automorphisms of order 15 and 21. [ABSTRACT FROM AUTHOR]

The distinguishing number D(G) of a graph G is the least integer d such that G has a vertex labeling with d labels that is preserved only by the trivial automorphism. A list assignment to G is an assignment L = {L(v)} v∈V (G) of lists of labels to the vertices of G. A distinguishing L-labeling of G is a distinguishing labeling of G where the label of each vertex v comes from L(v). The list distinguishing number of G, denoted by Dl(G), is the minimum k such that every list assignment to G in which |L(v)| = k for all v ∈ V (G) yields a distinguishing L-labeling of G. In this paper, we determine the list-distinguishing number for two families of graphs. We also characterize all graphs with the distinguishing number equal the list distinguishing number. Finally, we show that this characterization works for other list numbers of a graph. [ABSTRACT FROM AUTHOR]

In this paper, we present an innovative approach for the discovery of involutory maximum distance separable (MDS) matrices over finite fields F 2 q , derived from MDS self-dual codes, by employing a technique based on genetic algorithms. The significance of involutory MDS matrices lies in their unique properties, making them valuable in various applications, particularly in coding theory and cryptography. We propose a genetic algorithm-based method that efficiently searches for involutory MDS matrices, ensuring their self-duality and maximization of distances between code words. By leveraging the genetic algorithm's ability to evolve solutions over generations, our approach automates the process of identifying optimal involutory MDS matrices. Through comprehensive experiments, we demonstrate the effectiveness of our method and also unveil essential insights into automorphism groups within MDS self-dual codes. These findings hold promise for practical applications and extend the horizons of knowledge in both coding theory and cryptographic systems. [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]

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]

Complexity of sequences plays an important role in pseudorandom sequences and cryptography. In this paper, we present a construction of sequences with high nonlinear complexity from function fields. The main idea is to make use of function fields with many rational places as well as an automorphism of large order. We illustrate our construction through rational function fields and cyclotomic function fields in which there exist some automorphisms of large order. It turns out that we are able to: 1) slightly increase the length of the inversive sequence without losing nonlinear complexity and 2) obtain sequences with much larger nonlinear complexity than random sequences. [ABSTRACT FROM AUTHOR]

This paper surveys results concerning the quantization approach to the Jacobian Conjecture and related topics on noncommutative algebras. We start with a brief review of the paper and its motivations. The first section deals with the approximation by tame automorphisms and the Belov–Kontsevich Conjecture. The second section provides quantization proof of Bergman's centralizer theorem which has not been revisited for almost 50 years and formulates several related centralizer problems. In the third section, we investigate a free algebra analogue of a classical theorem of Białynicki-Birula's theorem and give a noncommutative version of this famous theorem. Additionally, we consider positive-root torus actions and obtain the linearity property analogous to the Białynicki-Birula theorem. In the last sections, we introduce Feigin's homomorphisms and we see how they help us in proving our main and fundamental theorems on screening operators and in the construction of our lattice W n -algebras associated with sl n , which is by far the simplest known approach concerning constructing such algebras until now. [ABSTRACT FROM AUTHOR]

In the present paper, we study the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of anabelian geometry. Let us recall that it is well-known that the natural homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the absolute Galois group of the given mixed-characteristic local field is injective. One main result of the present paper is that if the mixed-characteristic local field satisfies certain conditions, then the set of conjugates of the image of this injective homomorphism in the outer automorphism group is infinite, which thus implies that the image of this injective homomorphism is not normal in the outer automorphism group. In particular, one may conclude that it is impossible to establish a functorial group-theoretic reconstruction, from the absolute Galois group, of the "field-theoretic" subgroup, i.e., the image of this injective homomorphism, of the outer automorphism group. [ABSTRACT FROM AUTHOR]

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull 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. It has been shown that these algorithms are very effective in general if the size of the hull is small. The objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one-dimensional hull. It is shown that there are no such binary or ternary cyclic codes. Based on these characterizations, some constructions of linear codes with one-dimensional hull were given by employing quadratic number fields, partial difference sets, and difference sets. We also construct cyclic codes with one-dimensional hull. Some optimal codes with one-dimensional hull are obtained. [ABSTRACT FROM AUTHOR]

Let D be a nontrivial 3-(v, k, 1) design admitting a block-transitive group G of automorphisms. A recent work of Gan and the second author asserts that G is either affine or almost simple. In this paper, it is proved that if G is almost simple with socle an alternating group, then D is the unique 3-(10, 4, 1) design, and G = PGL (2 , 9) , M 10 or Aut (A 6) = S 6 : Z 2 , and G is flag-transitive. [ABSTRACT FROM AUTHOR]

All nonhyperbolic automorphisms of the 2-torus are not structurally stable, and it is generally impossible to predict the dynamics of their arbitrarily small perturbations. In this paper, given a representative of each algebraic conjugacy class of nonperiodic nonhyperbolic maps, a one-parameter family of diffeomorphisms is constructed, in which the zero value of the parameter corresponds to the given map and the nonzero values, to Morse–Smale diffeomorphisms. According to results of V. Z. Grines and A. N. Bezdenezhnykh, a Morse–Smale diffeomorphism of a closed orientable surface which induces a nonperiodic action on the fundamental group has nonempty heteroclinic set. It is proved that, in all of the constructed families, the diffeomorphisms corresponding to nonzero parameter values have nonempty orientable heteroclinic sets in which the number of orbits is determined by the automorphism being perturbed. [ABSTRACT FROM AUTHOR]

Representations of braid group B n on n ≥ 2 strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group V B n on n ≥ 2 strands is an extension of the classical braid group B n by the symmetric group S n . In this paper, we consider flat virtual braid groups F V B n on n ≥ 2 strands and construct a family of representations of F V B n by automorphisms of free groups of rank 2 n . It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case n = 2 and proved that the kernel contains a free group of rank two for n ≥ 3 . [ABSTRACT FROM AUTHOR]

This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank n, which is the free product Wn of n copies of Z/2Z. We prove that for n ≥ 5 the natural map Out(Wn) → Comm(Out(Wn)) is an isomorphism and that every isomorphism between finite index subgroups of Out(Wn) is given by a conjugation by an element of Out(Wn). [ABSTRACT FROM AUTHOR]

Group theory is the part of mathematics which addresses the study of symmetry. This paper extends the investigation of the soft group theory which Aktaş and Çağman have defined. We study new concepts in the soft group theory such as the center of the soft group, the kernel of soft homomorphism and soft automorphisms with their basic properties. Furthermore, the concept of soft point groups is introduced and the properties of these soft groups are studied. We state the concept of characteristic soft subgroups of a given soft group. Also, some theorems related to this concept are investigated. We study the characteristic soft subgroups of a given soft point group. The characteristic soft subgroups play a significant part in the study of the soft group theory and are useful for understanding the structure of a soft group and its soft automorphisms. As an application of characteristic soft subgroups, they allow us to identify and study important soft subgroups that are preserved under soft automorphisms. Also, practical applications for our theory can be conducted in future work such as the relation with other disciplines in sciences. [ABSTRACT FROM AUTHOR]

The topological symmetry group of an embedding Γ of an abstract graph γ in S 3 is the group of automorphisms of γ that can be realized by homeomorphisms of the pair (S 3 , Γ) . These groups are motivated by questions about the symmetries of molecules in space. The Petersen family of graphs is an important family of graphs for many problems in low-dimensional topology, so it is desirable to understand the possible groups of symmetries of their embeddings in space. In this paper, we find all the groups that can be realized as topological symmetry groups for each of the graphs in the Petersen family. Along the way, we also complete the classification of the realizable topological symmetry groups for K 3 , 3 . [ABSTRACT FROM AUTHOR]

In this paper, we characterize the q-Carleson measure for Dirichlet type spaces D p a on the unit ball when 0 < p < q < ¥ and a > 1 by using Carleson blocks and Carleson tubes. [ABSTRACT FROM AUTHOR]

Let H be an infinite-dimensional separable complex Hilbert space and B (H) the algebra of all bounded linear operators on H. In this paper, we characterized the linear maps ϕ : B (H) → B (H) , which are surjective up to compact operators preserving the set of left semi-Weyl operators in both directions. As an application, we proved that ϕ preserves the essential approximate point spectrum if and only if the ideal of all compact operators is invariant under ϕ and the induced map φ on the Calkin algebra is an automorphism. Moreover, we have i n d (ϕ (T)) = i n d (T) if both ϕ (T) and T are Fredholm. [ABSTRACT FROM AUTHOR]

Let K be the Galois field F q t of order q t , q = p e , p a prime, A = Aut (K) be the automorphism group of K and σ = (σ 0 , ... , σ d - 1) ∈ A d , d ≥ 1 . In this paper the following generalization of the Veronese map is studied: ν d , σ : ⟨ v ⟩ ∈ PG (n - 1 , K) ⟶ ⟨ v σ 0 ⊗ v σ 1 ⊗ ⋯ ⊗ v σ d - 1 ⟩ ∈ PG (n d - 1 , K). Its image will be called the (d , σ) -Veronese variety V d , σ . For d = t , σ a generator of Gal (F q t | F q) and σ = (1 , σ , σ 2 , ... , σ t - 1) , the (t , σ) -Veronese variety V t , σ is the variety studied in [9, 11, 13]. Such a variety is the Grassmann embedding of the Desarguesian spread of PG (n t - 1 , F q) and it has been used to construct codes [3] and (partial) ovoids of quadrics, see [9, 12]. Here, we will show that V d , σ is the Grassmann embedding of a normal rational scroll and any d + 1 points of it are linearly independent. We give a characterization of d + 2 linearly dependent points of V d , σ and for some choices of parameters, V p , σ is the normal rational curve; for p = 2 , it can be the Segre's arc of PG (3 , q t) ; for p = 3 V p , σ can be also a | V p , σ | -track of PG (5 , q t) . Finally, investigate the link between such points sets and a linear code C d , σ that can be associated to the variety, obtaining examples of MDS and almost MDS codes. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a suitable transformation from an element-based to a submodule-based interpretation of the traditional idea of transitivity in QTAG modules. We examine QTAG modules that are transitive in the sense that the module has an automorphism that sends one isotype submodule K onto any other isotype submodule K', unless this is impossible because either the submodules or the quotient modules are not isomorphic. Additionally, the classes of strongly transitive and strongly U-transitive QTAG modules are defined using a slight adaptations of this. This work investigates the latter class in depth, demonstrating that every α-module is strongly transitive with regard to countably generated isotype submodules. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study Alexander-type theorems in the case of smoothly bounded quasi-balanced domains of finite type in C 3 . Explicitly, the result that is proved in the paper is that the proper holomorphic self-maps of such domains are necessarily automorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we describe the periodic points of automorphisms of a one dimensional solenoid, considering it as the inverse limit, lim k (S1, k) of a sequence ( k) of maps on the circle S1. The periodic points are discussed for a class of automorphisms on some higher dimensional solenoids also. [ABSTRACT FROM AUTHOR]

Let G be a group and AutΦ(G) denote the group of all automorphisms of G centralizing G/Φ(G) elementwise. In this paper, we give a necessary and sufficient condition on a finite p-group G for the group AutΦ(G) to be elementary abelian. [ABSTRACT FROM AUTHOR]

Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters (45 , 22 , 10 , 11) , (49 , 24 , 11 , 12) , and (50 , 21 , 8 , 9) that have Z 6 as the automorphism group and constructed regular two-graphs from SRGs (45 , 22 , 10 , 11) , SRGs (49 , 24 , 11 , 12) , and SRGs (50 , 21 , 8 , 9) that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG (45 , 22 , 10 , 11) and 51 new regular two-graphs on 50 vertices leading to 398 new SRG (49 , 24 , 11 , 12) . [ABSTRACT FROM AUTHOR]

In this paper, we prove that all Coleman automorphisms of extension of an almost simple group by an abelian group or a simple group are inner. Using our methods we also show that the Coleman automorphisms of 2 -power order of an odd order group by an almost simple group are inner. In particular, these groups have the normalizer property. [ABSTRACT FROM AUTHOR]

We answer a question posed in a joint paper by Klyachko and Luneva about the optimality of an estimate for the cost of symmetry in graphs. The original estimate is that if one can delete vertices from a connected graph so that the resulting graph contains no connected subgraph isomorphic to , then in there exists a vertex subset of cardinality invariant under all automorphisms of such that the graph obtained from by deleting this subset contains no subgraph isomorphic to . We prove that there exists a graph for which this estimate is not optimal. [ABSTRACT FROM AUTHOR]

In this paper, we work over C . Criterions for an abelian surface to be products of elliptic curves are known. We give a new criterion from the view point of the Galois cover. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the actions of symmetric generalized (ϑ, η)-biderivations and generalized left (ϑ, η)-biderivations on Lie ideals and ideals of a prime ring A. It is shown that ℒ (nonzero square-closed Lie ideal of A) ⊆ ...(A), whenever traces of these derivations satisfy any of the following conditions: (i) ([l1,l2])Δ=0, (ii) (l1,l2)Δ ∈ ... (A), (iii) ([l1,l2])Δ = (l1)ϑ o (l2)Δ, (iv) (l1)Δ(l2)Δ + (l1)η(l2)ϑ ∈ ... (A), (v) a1((l1)Δ(l2)Δ + (l1l2)ϑ) = 0, (vi) (l1)Δ(l2)ϑ + (l1)ϑ(l2)Δ = 0, (vii) ([l1,l2])Δ + [(l1)Δ, l2] ∈ ... (A), (viii) (l1l2)Δ ± (l1)ϑ(l2)Δ + (l1l2)ϑ ∈ ... (A), ∀ l1, l2, ∈ ℒ, where 0 ≠ a1 ∈ A is a fixed element, Δ is a trace of these biadditive mappings and ϑ, η are automorphisms of A. [ABSTRACT FROM AUTHOR]

The token graphs of graphs have been studied at least from the 80’s with different names and by different authors. The Johnson graph J(n, k) is isomorphic to the k-token graph of the complete graph Kn. To our knowledge, the unique results about the automorphism groups of token graphs are for the case of the Johnson graphs. In this paper we begin the study of the automorphism groups of token graphs of another graphs. In particular we obtain the automorphism group of the k-token graph of the path graph Pn, for n 6= 2k. Also, we obtain the automorphism group of the 2-token graph of the following graphs: cycle, star, fan and wheel graphs. [ABSTRACT FROM AUTHOR]

The inner automorphism groups of quandles are related to the classification problem of quandles. The inner automorphism group of a quandle is generated by inner automorphisms which are presented by columns in the operation table of the quandle. In this paper, we describe inner automorphisms of an abelian extension of a quandle by expressing columns of the operation table of the extended quandle as columns of the operation table of the original quandle. Such a description will be helpful in studying inner automorphism groups of abelian extensions of quandles. [ABSTRACT FROM AUTHOR]

In this paper, we study the transfer of some alge-braic properties from the ring R to the ring of skew Hurwitz series (HR; !), where ! is an automorphism of R and vice versa. Different properties of skew Hurwitz series are studied with respect to various clean ring structures and semiclean ring structures. [ABSTRACT FROM AUTHOR]

In this paper we determine the automorphism groups of the profinite braid groups with four or more strings in terms of the profinite Grothendieck-Teichmuller group. [ABSTRACT FROM AUTHOR]

It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, Aut(Fn), has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of Aut(Fn) for n ≥ 4 where kth powers of all transvections act trivially for some fixed k. Thus, for some values of k, we deduce that the quotient of Aut(Fn) by the subgroup generated by kth powers of transvections contains nonabelian free groups. This expands on results of Malestein and Putman and of Bridson and Vogtmann. [ABSTRACT FROM AUTHOR]

In this paper, we make a start by considering the automorphisms of strong semilattice of π-groups, relating them to the automorphisms of underlying π-groups. We also provide a condition under which an automorphism of strong semilattice of π-groups can be constructed. [ABSTRACT FROM AUTHOR]

Let G be a finite group. The automorphism σ of a group G is said to be an absolute central automorphism, if for all x ∈ G, x-1xσ ∈ L(G), where L(G) be the absolute centre of G. In this paper, we study some properties of absolute central automorphisms of a given finite p-group. [ABSTRACT FROM AUTHOR]

All regular two-graphs having up to 36 vertices are known, and the first open case is the enumeration of two-graphs on 38 vertices. It is known that there are at least 191 regular two-graphs on 38 vertices and at least 18 regular two-graphs on 42 vertices. The number of descendants of these two-graphs is 6760 and 120, respectively. In this paper, we classify strongly regular graphs with parameters (41, 20, 9, 10) having nontrivial automorphisms and show that there are exactly 7152 such graphs. We enumerate all regular two-graphs on 38 and 42 vertices with at least one descendant whose full automorphism group is nontrivial and establish that there are at least 194 regular two-graphs on 38 vertices and at least 752 regular two-graphs on 42 vertices. Furthermore, we construct descendants with a trivial automorphism group of the newly constructed two-graphs and increase the number of known strongly regular graphs with parameters (37, 18, 8, 9) and (41, 20, 9, 10) to 6802 and 18439m respectively. This significantly increases the number of known strongly regular graphs with parameters (41, 20, 9, 10). [ABSTRACT FROM AUTHOR]

We consider the class of generalized algebraic geometry codes (GAG codes) formed by two collections of places, with places of the same degree in each collection. We introduce the concept of N 1 N 2 -automorphism group of a GAG code in this class-that is, a subgroup of the automorphism group of the code. Then we determine a subgroup of the N 1 N 2 -automorphism group in the general case and the N 1 N 2 -automorphism group itself in the rational function field case. We also explicitly construct such a group. This paper presents a method to obtain similar results for the GAG codes that have more collections of places of the same degree in their construction. [ABSTRACT FROM AUTHOR]

We extend Choiceless Polynomial Time (CPT), the currently only remaining promising candidate in the quest for a logic capturing Ptime, so that this extended logic has the following property: for every class of structures for which isomorphism is definable, the logic automatically captures Ptime. For the construction of this logic, we extend CPT by a witnessed symmetric choice operator. This operator allows for choices from definable orbits. But, to ensure polynomial-time evaluation, automorphisms have to be provided to certify that the choice set is indeed an orbit. We argue that, in this logic, definable isomorphism implies definable canonization. Thereby, our construction removes the non-trivial step of extending isomorphism definability results to canonization. This step was a part of proofs that show that CPT or other logics capture Ptime on a particular class of structures. The step typically required substantial extra effort. [ABSTRACT FROM AUTHOR]

In the paper, automorphisms are studied for free groups of varieties given by a family of identities in the well-known infinite independent system of identities involving two variables that was constructed by S. I. Adian to solve the finite basis problem in group theory. It is proved that every normal automorphism (i.e., an automorphism that stabilizes any normal subgroup) of noncyclic free groups of these varieties is an inner automorphism. [ABSTRACT FROM AUTHOR]

A lattice of definability subspaces in the order of rational numbers is described. It is proved that this lattice consists of five subspaces defined in the paper that are generated by the following relations: "equality," "less," "between," "cycle," and "linkage." For each of the subspaces, its width (the minimum number of arguments of a generating relation) is found and a convenient description of the automorphism group is given. Although the structure of this lattice was known previously, the proof in the paper is of syntactic nature and avoids the use of a group-theoretical method. [ABSTRACT FROM AUTHOR]