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In this paper, we classified Hadamard 2-(51, 25, 12) designs having a non-abelian automorphism group of order 10, i.e., Frob10 ≌ Z5: Z2, where a subgroup isomorphic to Z2 fixed setwise all the Z5-orbits of the sets of points and blocks. Furthermore, we classified 2-(59, 29, 14) designs having a non-abelian automorphism group of order 14, i.e., Frob14 ≌ Z7: Z2, where a subgroup isomorphic to Z2 fixed setwise all the Z7-orbits of the sets of points and blocks. We also showed that there was no Hadamard 2-(59, 29, 14) design with automorphism group Frob21 ≌ Z7: Z3, where a subgroup of order 3 fixed setwise all the Z7-orbits of points and blocks. Additionally, we used Hadamard 2-(59, 29, 14) designs obtained to construct ternary linear codes and linear codes over the finite field of order 5. We constructed ternary self-dual codes and self-dual codes over the finite field of order 5 from the corresponding Hadamard matrices of order 60. [ABSTRACT FROM AUTHOR]

The symplectic group S p 2 n (2) has an affine maximal subgroup of structure A S p n = 2 2 n - 1 : S p 2 n - 2 (2) which is a split extension of an elementary abelian 2-group N = 2 2 n - 1 by G = S p 2 n - 2 (2) . The vector space N = 2 2 n - 1 and its dual N ∗ are not equivalent as 2 n - 1 dimensional G-modules over GF(2). Therefore, a split extension of the form G ¯ n = N ∗ : S p 2 n - 2 (2) ≇ N : S p 2 n - 2 (2) exists. In this paper, it will be shown that G ¯ n ≅ Aut (2 2 n - 2 : S p 2 n - 2 (2)) = 2 2 n - 2 : S p 2 n - 2 (2) : 2 for n ≥ 3 . Moreover, the ordinary irreducible characters of G ¯ n are studied through the lens of Fischer-Clifford theory. As an example, the Fischer-Clifford matrix technique is used to construct the set Irr (G ¯ 5) of the group G ¯ 5 = 2 9 : S p 8 (2) which is associated with the affine subgroup A S p 5 = 2 9 : S p 8 (2) of S p 10 (2) . [ABSTRACT FROM AUTHOR]

The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, as well as the mathematical structure of the automorphism group, which captures all of the symmetries of a graph. The paper also discusses the significance of groups in Abstract Algebra and their relevance to understanding the behavior of mathematical systems. Overall, the findings offer an inclusive understanding of the relationship between symmetry groups and automorphism groups, paving the way for further research in this area. [ABSTRACT FROM AUTHOR]

Given a finite field F q and a positive integer n , a flag is a sequence of nested F q -subspaces of a vector space F q n and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes containing all the subspaces of prescribed dimensions that form the flags in the flag code. In this paper we address the notion of equivalence for flag codes and explore in which situations such an equivalence can be reduced to the equivalence of the corresponding projected codes. In addition, this study leads to new results concerning the automorphism group of certain families of flag codes, some of them also introduced in this paper. [ABSTRACT FROM AUTHOR]

This paper presents a new characterisation of the Fermat curve, according to the arrangement of Galois points. [ABSTRACT FROM AUTHOR]

In this paper, we characterize weakly pseudoconvex domains of finite type in ℂn in terms of the boundary behavior of automorphism orbits by using the scaling method. [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]

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]

We investigate the automorphism group of the substructure ordering of finite directed graphs. The second author conjectured that it is isomorphic to the 768-element group (Z 2 4 × S 4) ⋊ α Z 2 . Though unable to prove it, we solidify this conjecture by showing that the automorphism group behaves as expected by the conjecture on the first few levels of the poset in question. With the use of computer calculation we analyze the first four levels holding 3160 directed graphs. [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]

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with n vertices and of valence d , d ≤ 4 , is at most c d n where c 3 = 1 and c 4 = 9. Whether such a constant c d exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism g of a finite connected 3-valent vertex-transitive graph Γ, Γ ≇ K 3 , 3 , has a regular orbit, that is, an orbit of 〈 g 〉 of length equal to the order of g. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of g. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms C of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of C. [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. Denote by L (G) the absolute center of a group G, that is the set of elements of G fixed by all automorphisms of G. By using the structure of the automorphism group of a ZM-group, in this paper we prove that cyclic groups C N , N ∈ N * , are completely L-realisable. [ABSTRACT FROM AUTHOR]

A graph X $X$ is said to be unstable if the direct product X×K2 $X\times {K}_{2}$ (also called the canonical double cover of X $X$) has automorphisms that do not come from automorphisms of its factors X $X$ and K2 ${K}_{2}$. It is nontrivially unstable if it is unstable, connected, nonbipartite, and distinct vertices have distinct sets of neighbours. In this paper, we prove two sufficient conditions for stability of graphs in which every edge lies on a triangle, revising an incorrect claim of Surowski and filling in some gaps in the proof of another one. We also consider triangle‐free graphs, and prove that there are no nontrivially unstable triangle‐free graphs of diameter 2. An interesting construction of nontrivially unstable graphs is given and several open problems are posed. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra is the semidirect product of the special linear Lie algebra s l 2 and the Heisenberg algebra h n . In this paper, we describe explicitly the structure of the derivation algebra and automorphism group of the n-th Schrödinger algebra. Especially, the automorphism group of the Schrödinger algebra is obtained. [ABSTRACT FROM AUTHOR]

In this paper we compute the automorphism group of the curves X a , b , n , s and Y n , s introduced in Tafazolian et al. [27] as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained. [ABSTRACT FROM AUTHOR]

Let D = (P , B) be a symmetric 2- (v , k , λ) design admitting a flag-transitive, point-imprimitive automorphism group G that leaves invariant a non-trivial partition Σ of P. Praeger and Zhou [42] have shown that, there is a constant k 0 such that, for each B ∈ B and Δ ∈ Σ , the size of | B ∩ Δ | is either 0 or k 0. In the present paper we show that, if k > λ (λ − 3) / 2 and k 0 ⩾ 3 , D is isomorphic to one of the known flag-transitive, point-imprimitive symmetric 2-designs with parameters (45 , 12 , 3) or (96 , 20 , 4). [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic p > 0. A pressing problem in the theory of algebraic curves is the determination of the p -rank of a (nonsingular, projective, irreducible) curve X over K. This birational invariant affects arithmetic and geometric properties of X , and its fundamental role in the study of the automorphism group Aut (X) has been noted by many authors in the past few decades. In this paper, we provide an extensive study of the p -rank of curves of Fermat type y m = x n + 1 over K = F ¯ p. We determine a combinatorial formula for this invariant in the general case and show how this leads to explicit formulas of the p -rank of several such curves. By way of illustration, we present explicit formulas for more than twenty subfamilies of such curves, where m and n are generally given in terms of p. We also show how the approach can be used to compute the p -rank of other types of curves. [ABSTRACT FROM AUTHOR]

In this paper, we classified Hadamard $ 2 $-$ (51, 25, 12) $ designs having a non-abelian automorphism group of order 10, i.e., $ {\rm{Frob}}_{10} \cong Z_5:Z_2 $, where a subgroup isomorphic to $ Z_2 $ fixed setwise all the $ Z_5 $-orbits of the sets of points and blocks. Furthermore, we classified $ 2 $-$ (59, 29, 14) $ designs having a non-abelian automorphism group of order 14, i.e., $ {\rm{Frob}}_{14} \cong Z_7:Z_2 $, where a subgroup isomorphic to $ Z_2 $ fixed setwise all the $ Z_7 $-orbits of the sets of points and blocks. We also showed that there was no Hadamard $ 2 $-$ (59, 29, 14) $ design with automorphism group $ {\rm{Frob}}_{21} \cong Z_7:Z_3 $, where a subgroup of order 3 fixed setwise all the $ Z_7 $-orbits of points and blocks. Additionally, we used Hadamard $ 2 $-$ (59, 29, 14) $ designs obtained to construct ternary linear codes and linear codes over the finite field of order 5. We constructed ternary self-dual codes and self-dual codes over the finite field of order 5 from the corresponding Hadamard matrices of order 60.

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]

Given an affine algebraic variety X, we prove that if the neutral component \mathrm {Aut}^\circ (X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G contains a closed connected nested ind-subgroup H\subset G, and for any g\in G some positive power of g belongs to H, then G=H. [ABSTRACT FROM AUTHOR]

Let D (R m , n (q)) be the Drinfeld double of Radford Hopf algebra R m , n (q) and D (H s , t) be the Drinfeld double of generalized Taft algebra H s , t . Both D (R m , n (q)) and D (H s , t) have very symmetric structures. We calculate all Hopf automorphisms of D (R m , n (q)) and D (H s , t) , respectively. Furthermore, we prove that the Hopf automorphism group A u t H o p f (D (R m , n (q))) is isomorphic to the direct sum Z n ⨁ Z m of cyclic groups Z m and Z n , the Hopf automorphism group A u t H o p f (D (H s , t)) is isomorphic to the semi-direct products k * ⋊ Z d of multiplicative group k * and cyclic group Z d , where s = t d , k * = k \ { 0 } , and k is an algebraically closed field with char (k) ∤ t . [ABSTRACT FROM AUTHOR]

In this paper, we generalize the construction of Deligne–Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate some properties of such generalized Deligne–Hitchin twistor space as a complex analytic manifold. More precisely, we show it admits holomorphic sections whose normal bundle contains a semistable subbundle with positive degree and whose energy is semi-negative, and it carries a balanced metric. Moreover, we also study the automorphism groups of the Hodge moduli spaces and the generalized Deligne–Hitchin twistor spaces. [ABSTRACT FROM AUTHOR]

A graph is symmetric if its automorphism group is transitive on the arcs of the graph. Guo et al. determined all of the connected seven-valent symmetric graphs of order 8 p for each prime p. We shall generalize this result by determining all of the connected seven-valent symmetric graphs of order 8 p q with p and q to be distinct primes. As a result, we show that for each such graph of Γ , it is isomorphic to one of seven graphs. [ABSTRACT FROM AUTHOR]

In a recent paper (Araya and Harada, 2023), Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for 145 distinct values of the number A12 of codewords of minimum weight 12, and raised the question about the existence of codes for other values of A12. In this note, we use symmetric 2-(47,23,11) designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for 150 new values of A12.

The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, as well as the mathematical structure of the automorphism group, which captures all of the symmetries of a graph. The paper also discusses the significance of groups in Abstract Algebra and their relevance to understanding the behavior of mathematical systems. Overall, the findings offer an inclusive understanding of the relationship between symmetry groups and automorphism groups, paving the way for further research in this area.

It is known that Z p s -linear codes, which are the Gray map image of Z p s -additive codes (linear codes over Z p s ), are systematic and a systematic encoding has been found. This makes Z p s -linear codes suitable to apply the permutation decoding method. This technique is also based on the existence of r -PD-sets, which are subsets of the permutation automorphism group of the code. In this paper, we study the permutation automorphism group of Z p s -linear generalized Hadamard codes of type (n ; t 1 , ... , t s) and show how to construct r -PD-sets of size r + 1 , for all r up to an upper bound, in order to be able to perform a partial permutation decoding for these codes. [ABSTRACT FROM AUTHOR]

Let G be a graph. The 2- token graph F 2 (G) of G is with vertex set all the 2-subsets of V (G) such that two 2-subsets are adjacent if their symmetric difference is exactly an edge of G. In this paper, the full automorphism group of the 2-token graph of the Hamming graph is determined. [ABSTRACT FROM AUTHOR]

Let S be a non-empty finite set. A flag of S is a set f of non-empty proper subsets of S such that X ⊆ Y or Y ⊆ X for all X , Y ∈ f. The set { | X | : X ∈ f } is called the type of f. Two flags f and f ′ are in general position with respect to S if X ∩ Y = ∅ or X ∪ Y = S for all X ∈ f and Y ∈ f ′. For a fixed type T , Klaus Metsch defined the general position graph Γ (S , T) whose vertices are the flags of S of type T with two vertices being adjacent when the corresponding flags are in general position. In this paper, we characterize the full automorphism groups of Γ (S , T) in the case that | T | = 2. In particular, we solve an open problem proposed by Klaus Metsch. [ABSTRACT FROM AUTHOR]

We give a condition on the defining graph of a right-angled Artin group, which implies that its automorphism group is virtually indicable, that is, it has a finite index subgroup that admits a homomorphism onto Z. We use this as a part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence, we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group. [ABSTRACT FROM AUTHOR]