Search

In this paper, we give the complete classification of 5-dimensional complex solvable symmetric Leibniz algebras. [ABSTRACT FROM AUTHOR]

AbstractLet FI(X, K) be the finitary incidence algebra of a non-connected partially ordered set X over a field K of characteristic different from 2. For the case where every multiplicative automorphism of FI(X, K) is inner, we present necessary and sufficient conditions for two involutions on FI(X, K) to be equivalent. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G), f a permutation of V (G). Define δf(x,y) = |d(x,y) − d(f(x),f(y))| and δf(G) =∑ δf(x,y), where the sum is taken over all unordered pairs x, y of distinct vertices of G. Let π(G) denote the smallest positive value of δf(G) among all permutations f of V (G). A permutation f with δf(G) = π(G) is called a near automorphism of G. In this paper, the near automorphisms of the complement or the square of a cycle are characterized. Moreover, π(Cn¯) and π Cn2 are determined. [ABSTRACT FROM AUTHOR]

We present the following constructions: extension of a trivial quandle by a group with a nontrivial abelian subgroup, a quandle over a noncommutative group by an arbitrary antiautomorphism, a quandle presenting from a generalized Alexander quandle with the replacement of the automorphism by a central antiautomorphism, and a quandle over an -dimensional module depending on parameters with in a commutative unital ring. [ABSTRACT FROM AUTHOR]

AbstractAn additive action on an irreducible algebraic variety X is an effective action with an open orbit of the vector group . Any two additive actions on X are conjugate by a birational automorphism of X. We prove that, if X is the projective space, the conjugating element can be chosen in the affine Cremona group and it is given by so-called basic polynomials of the corresponding local algebra. [ABSTRACT FROM AUTHOR]

Let A be a finite nilpotent group acting fixed point freely on the finite (solvable) group G by automorphisms. It is conjectured that the nilpotent length of G is bounded above by ℓ (A) , the number of primes dividing the order of A counted with multiplicities. In the present paper we consider the case A is cyclic and obtain that the nilpotent length of G is at most 2 ℓ (A) if | G | is odd. More generally we prove that the nilpotent length of G is at most 2 ℓ (A) + c (G ; A) when G is of odd order and A normalizes a Sylow system of G where c (G ; A) denotes the number of trivial A -modules appearing in an A -composition series of G. [ABSTRACT FROM AUTHOR]

A cycle in a graph is consistent if the automorphism group of the graph admits a one‐step rotation of this cycle. A thorough description of consistent cycles of arc‐transitive subgroups in the full automorphism groups of finite cubic symmetric graphs is given. [ABSTRACT FROM AUTHOR]

We consider linear isotopes of commutative groups, i.e., central quasigroups and study the invertibility and orthogonality conditions. It turns out that it is sufficient to study these conditions solely for the unitary isotopes, i.e., for isotopes, which have an idempotent. We establish criteria for the possession of each invertibility property (inverse property, crossed inverse property, and mirroring) for unitary central and matrix quasigroups. In particular, for matrices of the second order, we describe the corresponding matrix quasigroups over the fields of characteristics 2 and 3. The orthogonality criteria are obtained for matrix quasigroups with the indicated invertibility properties. [ABSTRACT FROM AUTHOR]

Let M be the free metabelian Leibniz algebra of rank two over a field K of characteristic 0. In this article, we show that an endomorphism of M which preserves the orbit of a nontrivial certain kind of element w ∈ M ′ is an automorphism. Additionally, we give some test elements. [ABSTRACT FROM AUTHOR]

We prove analogs of A. Selberg's result for finitely generated subgroups of \operatorname {Aut}(A) and of Engel's theorem for subalgebras of \operatorname {Der}(A) for a finitely generated associative commutative algebra A over an associative commutative ring. We prove also an analog of the theorem of W. Burnside and I. Schur about local finiteness of torsion subgroups of \operatorname {Aut}(A). [ABSTRACT FROM AUTHOR]

In this paper, we prove that for any k ≥ 3 , there exist infinitely many minimal asymmetric k -uniform hypergraphs. This is in a striking contrast to k = 2 , where it has been proved recently that there are exactly 18 minimal asymmetric graphs. We also determine, for every k ≥ 1 , the minimum size of an asymmetric k -uniform hypergraph. [ABSTRACT FROM AUTHOR]

In this paper, we describe the split exact sequences of skew left braces. We define a free action of the second cohomology group of a skew left brace H by Ann (I) on Ext α (H , I) and show that this action becomes transitive if I is a trivial skew brace. We also generalize the Well's type exact sequence for extensions by the trivial skew brace. [ABSTRACT FROM AUTHOR]

Let K be a field and X a connected partially ordered set. In the first part of this paper, we show that the finitary incidence algebra FI(X, K) of X over K has an involution of the second kind if and only if X has an involution and K has an automorphism of order 2. We also give a characterization of the involutions of the second kind on FI(X, K). In the second part, we give necessary and sufficient conditions for two involutions of the second kind on FI(X, K) to be equivalent in the case where char K ≠ 2 and every multiplicative automorphism of FI(X, K) is inner. [ABSTRACT FROM AUTHOR]

Let Q = (a,b / R) denote the quaternion algebra over the reals which is by the Frobenius Theorem either split or the division algebra H of Hamilton's quaternions. We have presented explicitly in [4] the matrix of a typical derivation of Q. Given a derivation d ∈ Der(H), we show that the matrix D ∈ Ma (ℝ) that represents d on the linear subspace ... of pure quaternions provides a pair of idempotent matrices AdjD and - D² that correspond bijectively to the involutary matrix Ʃof a quaternion involution σ and present several equations involving these matrices. In particular, we deal with commuting derivations of H and introduce some results to guarantee commutativity. We also mention briefly eigenspace decomposition of a derivation. [ABSTRACT FROM AUTHOR]

We call an algebra A commutator-simple if [A,A] does not contain nonzero ideals of A. After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation D\colon L^1(G)\to L^1(G), where G is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras. [ABSTRACT FROM AUTHOR]

This article extends Bayer-Fluckiger's theorem on characteristic polynomials of isometries on an even unimodular lattice to the case where the isometries have determinant −1. As an application, we show that the logarithm of every Salem number of degree 20 is realized as the topological entropy of an automorphism of a nonprojective K3 surface. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let κ (G) denote the probability that two randomly selected elements of G are conjugate. Two elements of G are called autoconjugate if one is the image of other under an automorphism of G. Let P a (G) denote the probability that two randomly selected elements of G are autoconjugate. We calculate P a (G) for some finite groups and some lower bounds of P a (G) . We also prove that P a (G) = κ (G) if and only if Aut (G) = Aut c (G) , where Aut c (G) denotes the group of all class-preserving automorphisms of G. This gives another perspective on a question of Avinoam Mann. [ABSTRACT FROM AUTHOR]

The aim of the paper is to give an explicit description of bi-quadratic algebras on 3 generators with PBW basis. [ABSTRACT FROM AUTHOR]

The automorphism group of a K3 surface with Picard number two is either the infinite cyclic group or the infinite dihedral group, if it is infinite. In this paper, we determine some conditions for a K3 surface of Picard number two to have the infinite dihedral automorphism group. [ABSTRACT FROM AUTHOR]

Let F be a free Leibniz algebra generated by the set X = { x 1 , ... , x n } over the field K of characteristic 0 and let R be an ideal of F. In this study, a necessary and sufficient condition for n elements of the Leibniz algebra F / R ′ to be a generating set is given. [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 풜 of the polynomial algebra K[x] that contains the ideal xmK[x] for some m ≥ 1. It is proven that the set 풜 =∐m,Γ풜(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 풜(m, Γ) is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on 풜(m, Γ). An isomorphism criterion is given for the algebras in 풜. For each algebra A ∈풜(m, Γ), explicit sets of generators and defining relations are given and the automorphism group AutK(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 |AutK(A)| < dim K(A/픠A) where 픠A is the conductor of A. The set of orders of the automorphism groups of the algebras in 풜(m, Γ) is explicitly described. [ABSTRACT FROM AUTHOR]

Let ℧ be a prime ring of char(℧) ≠ 2 with its center Z. This article introduces new classes of endomorphisms and investigates how they relate to antiautomorphisms of prime rings and the commutativity of prime rings. Additionally, we fully describe and classify some of these endomorphisms. We also give examples to prove the necessity of the numerous restrictions included in the hypotheses of our results. [ABSTRACT FROM AUTHOR]

Let ℒ be a bounded lattice and α : ℒ → ℒ be its automorphism. In this paper, we study zero‐divisor graph of ℒ with respect to an automorphism α. It is a simple undirected graph and denoted by Γα(ℒ). Some combinatorial structures such as coloring, diameter and girth were given for Γα(ℒ). [ABSTRACT FROM AUTHOR]

This paper is devoted to the descriptions of automorphisms and local automorphisms on complex solvable Lie algebras of maximal rank. First, it is established that any automorphism on a solvable Lie algebra of maximal rank can be represented as a product (composition) of inner, diagonal and graph automorphisms. We apply the description of automorphism to the specification of automorphisms on solvable Lie algebras of maximal rank with abelian nilradical, and to the description of automorphisms of standard Borel subalgebras of complex simple Lie algebras. Based on the representation of an automorphism, it is proved that all local automorphisms on a solvable Lie algebra of maximal rank are global automorphisms. We also present two examples of solvable Lie algebras which are not of maximal rank and have different behaviours of local automorphisms. Namely, the first algebra does not admit pure local automorphisms, while the second algebra admits a local automorphism which is not an automorphism. [ABSTRACT FROM AUTHOR]

This article examines the commutativity of rings with antiautomorphisms, specifically when they are equipped with derivations that satisfy certain algebraic identities. Moreover, we present examples to demonstrate the necessity of the various restrictions imposed in the hypotheses of our theorems. [ABSTRACT FROM AUTHOR]

For any real \beta let H^2_\beta be the Hardy-Sobolev space on the unit ball \mathbb {B}_{n}, n\geq 1. H^2_\beta is a reproducing kernel Hilbert space and its reproducing kernel is bounded when \beta >n/2. In this paper, we characterize when the composition operator C_{\varphi } on H^{2}_{\beta } is Fredholm for a non-constant analytic map \varphi :\mathbb {B}_{n}\to \mathbb {B}_{n}. [ABSTRACT FROM AUTHOR]

Let X be a hyperelliptic algebraic curve and let M (E 6) be the moduli space of polystable principal E 6 -bundles over X. Suppose, in addition, that the outer involution σ of E 6 acts as the hyperelliptic involution of X. Then, an automorphism of M (E 6) is defined which acts by E ↦ σ ∗ (E ∗) , where E is a principal E 6 -bundle over X seen as a vector bundle through the fundamental irreducible 27-dimensional representation of E 6 . In this paper, Galois E 6 -bundles over X are defined and related to the fixed points of the above automorphism of M (E 6) . If P E 6 is the centerless group with Lie algebra e 6 , then Galois P E 6 -bundles over X are also defined and related to Galois E 6 -bundles. Finally, a specific expression for a certain family of Galois E 6 -bundles over X is given and some implications of the study in terms of representations of the fundamental group π 1 (X) of the base curve are drawn. [ABSTRACT FROM AUTHOR]

Ivanov introduced the shape of a Majorana algebra as a record of the 2-generated subalgebras arising in that algebra. As a broad generalisation of this concept and to free it from the ambient algebra, we introduce the concept of an axet and shapes on an axet. A shape can be viewed as an algebra version of a group amalgam. Just like an amalgam, a shape leads to a unique algebra completion which may be non-trivial or it may collapse. Then for a natural family of shapes of generalised Monster type we classify all completion algebras and discover that a great majority of them collapse, confirming the observations made in an earlier paper [12]. [ABSTRACT FROM AUTHOR]

Abstract Let X be a compact algebraic curve of genus g ≥ 2. The nontrivial outer automorphism σ of the complex Lie group E 6 acts on the moduli space M(E 6) of principal E 6-bundles over X , and this action defines an automorphism fσ of M(E 6). The group H 1(X, Z (E 6)) of principal Z(E 6)-bundles over X also acts on M(E 6) by tensor product, Z(E 6) being the center of E 6, so each choice of an element L ∈ H 1(X, Z(E 6)) defines an automorphism fL of M(E 6). In this paper two theorems describing the simple fixed points of the automorphism fL of M(E 6) and the composition fL ◦fσ are proved. [ABSTRACT FROM AUTHOR]

The aim of the paper is to introduce a new class of rings — the inner generalized Weyl algebras (IGWA) — and to give simplicity criteria for them. For each IGWA A a derivative series of IGWAs, A → A′→ A″ →⋯ → A(α) →⋯, is attached where α is an arbitrary ordinal. In general, all rings A(α) are distinct. A new construction of rings, the inner (σ,τ,a)-extension of a ring, is introduced (where σ and τ are endomorphisms of a ring D and a ∈ D). [ABSTRACT FROM AUTHOR]

Let K be an uncountable algebraically closed field of characteristic zero. An affine algebraic variety X over K is toral if it is isomorphic to a closed subvariety of a torus (K*)d. We study the group Aut(X) of regular automorphisms of a toral variety X. We prove that if T is a maximal torus in Aut(X), then X is a direct product Y × T, where Y is a toral variety with a trivial maximal torus in the automorphism group. We show that knowing Aut(Y), one can compute Aut(X). In the case when the rank of the group K[Y]*/K* is dim Y + 1, the group Aut(Y) is described explicitly. [ABSTRACT FROM AUTHOR]

We characterize the sets of homomorphisms, endomorphisms and automorphisms of n-ary groups with cyclic retracts. [ABSTRACT FROM AUTHOR]

Let P ( sl 2 (K)) be the Poisson enveloping algebra of the Lie algebra sl 2 (K) over an algebraically closed field K of characteristic zero. The quotient algebras P ( sl 2 (K)) / (C P − λ) , where C P is the standard Casimir element of sl 2 (K) in P ( sl 2 (K)) and 0 ≠ λ ∈ K , are proven to be simple in [U. Umirbaev and V. Zhelyabin, A Dixmier theorem for Poisson enveloping algebras, J. Algebra 568 (2021) 576–600]. Using a result by Makar–Limanov [22], we describe generators of the automorphism group of P ( sl 2 (K)) / (C P − λ) and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by Dixmier [Quotients simples de l'algebre enveloppante de 2 , J. Algebra 24 (1973) 551–564] and O. Fleury [Sur les sous-groupes finis de Aut U ( 2) et Aut U () , J. Algebra 200 (1998) 404–427] for the quotient algebras U ( sl 2 (K)) / (C U − λ) , where C U is the standard Casimir element of sl 2 (K) in the universal enveloping algebra U ( sl 2 (K)) , we prove that the automorphism groups of P ( sl 2 (K)) / (C P − λ) and U ( sl 2 (K)) / (C U − λ) are isomorphic. [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]

Let f be an automorphism of a complex Enriques surface Y and let pf denote the characteristic polynomial of the isometry f* of the numerical Néron--Severi lattice of Y induced by f. We combine a modification of McMullen's method with Borcherds' method to prove that the modulo-2 reduction (pf (x) mod 2) is a product of modulo-2 reductions of (some of) the five cyclotomic polynomials Φm, where m ≤ 9 and m is odd. We study Enriques surfaces that realize modulo-2 reductions of Φ7, Φ9 and show that each of the five polynomials (Φm(x) mod 2) is a factor of the modulo-2 reduction (pf (x) mod 2) for a complex Enriques surface. [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]

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]

We prove the endomorphism conjecture for graded posets with largest Whitney number at most 4. [ABSTRACT FROM AUTHOR]

In this article, we treat the space H of real quaternions as a Lie algebra equipped with its commutator product. We show that all involutions of this Lie algebra that are automorphisms (respectively, anti-automorphisms) and restrict to the identity on the centre R ⋅ 1 (sometimes called automorphisms of the first kind) are actually algebra automorphisms (resp. anti-automorphisms) of the division algebra of quaternions, which we characterized in an earlier paper. If we compose with scalar multiplication by − 1 , we obtain all involutive automorphisms and anti-automorphisms of the second kind, i.e. those for which the centre is contained in the − 1 -eigenspace. Together, we have a complete determination of all involutive (anti-)automorphisms on the quaternionic Lie algebra L (H). From this determination of all the involutive (anti-)automorphisms of L (H) , one can identify via a standard bijective correspondence all the involutive (anti-)automorphisms for the corresponding simply connected multiplicative quaternion Lie group H − { 0 }. We carry out this determination explicitly. [ABSTRACT FROM AUTHOR]

Automorphism groups of formal matrix algebras with zero trace ideals are studied. Such an algebra is represented as a splitting extension of some ring by some nilpotent ideal. Using this extension, the study of the structure of the automorphism group of an algebra in a certain sense is reduced to the study of the structure of some its subgroups and quotient groups. Then the structure of these subgroups and factor groups is found. The case of formal triangular matrix rings is specially considered. At the end of the paper we study automorphisms of ordinary matrix rings. In this case, a well-known homomorphism from the group of outer automorphisms to the Picard group of a certain ring is used. [ABSTRACT FROM AUTHOR]

Let A be a free abelian group of rank n (n ≥ 2) and ϕ be an automorphism of A, then A ⋊ 〈 ϕ 〉 is a polycyclic group. This paper investigates the residual finiteness of A ⋊ 〈 ϕ 〉 , and the special case that ϕ is a cyclic permutation of the direct components twisted by a rational integer m. [ABSTRACT FROM AUTHOR]

Assume that a Lie type algebra admits a Frobenius group of automorphisms with cyclic kernel of order and complement of order such that the fixed-point subalgebra with respect to is trivial and the fixed-point subalgebra with respect to is nilpotent of class . If the ground field contains a primitive th root of unity, then the algebra is nilpotent and the nilpotency class is bounded in terms of and . The result extends the well-known theorem of Khukhro, Makarenko, and Shumyatsky on Lie algebras with metacyclic Frobenius group of automorphisms. [ABSTRACT FROM AUTHOR]

Let f be an automorphism of a complex Enriques surface Y and let pf denote the characteristic polynomial of the isometry f* of the numerical Néron-Severi lattice of Y induced by f. We combine a modification of McMullen's method with Borcherds' method to prove that the modulo-2 reduction (pf (x) mod 2) is a product of modulo-2 reductions of (some of) the five cyclotomic polynomials Φm, where m ≤ 9 and m is odd. We study Enriques surfaces that realize modulo-2 reductions of Φ7, Φ9 and show that each of the five polynomials (Φm(x) mod 2) is a factor of the modulo-2 reduction (pf (x) mod 2) for a complex Enriques surface. [ABSTRACT FROM AUTHOR]

Suppose that a finite group G admits a soluble group of coprime automorphisms A. We prove that if, for some positive integer m, every element of the centralizer CG(A) has a left Engel sink of cardinality at most m (or a right Engel sink of cardinality at most m), then G has a subgroup of (|A|,m)-bounded index which has Fitting height at most 2α(A) + 2, where α(A) is the composition length of A. We also prove that if, for some positive integer r, every element of the centralizer CG(A) has a left Engel sink of rank at most r (or a right Engel sink of rank at most r), then G has a subgroup of (|A|, r)-bounded index which has Fitting height at most 4α(A) + 4α(A) + 3. Here, a left Engel sink of an element g of a group G is a set 픈 (g) such that for every x ∈ G all sufficiently long commutators [...[[x, g], g],... , g] belong to 픈 (g). (Thus, g is a left Engel element precisely when we can choose (g) = {1}.) A right Engel sink of an element g of a group G is a set ℜ(g) such that for every x ∈ G all sufficiently long commutators [...[[g, x], x],... , x] belong to ℜ(g). Thus, g is a right Engel element precisely when we can choose ℜ(g) = {1}. [ABSTRACT FROM AUTHOR]

We prove that every derivation and every locally nilpotent derivation of the subalgebra K [ x n , x n − 1 y , ... , x y n − 1 , y n ] , where n ≥ 2 , of the polynomial algebra K [ x , y ] in two variables over a field K of characteristic zero is induced by a derivation and a locally nilpotent derivation of K [ x , y ] , respectively. Moreover, we prove that every automorphism of K [ x n , x n − 1 y , ... , x y n − 1 , y n ] over an algebraically closed field K of characteristic zero is induced by an automorphism of K [ x , y ]. We also show that the group of automorphisms of K [ x n , x n − 1 y , ... , x y n − 1 , y n ] admits an amalgamated free product structure. [ABSTRACT FROM AUTHOR]

In this paper a relation between completely semiprime ideals of a ring R and those of R[x; δ] has been given; δ is a derivation of R. It has been proved that if P is a completely semiprime ideal of R such that δ(P) ⊆ P, then P[x; δ] is a completely semiprime ideal of R[x; δ]. [ABSTRACT FROM AUTHOR]

A totally symmetric set is a subset of a group such that every permutation of the subset can be realized by conjugation in the group. The (non-)existence of large totally symmetric sets obstruct homomorphisms, so bounds on the sizes of totally symmetric sets are of particular use. In this paper, we prove that if a group has a totally symmetric set of size fc, it must have order at least (k + 1)!. We also show that with three exceptions, {(1 i) | i = 2...,n} C Sn is the only totally symmetric set making this bound sharp, a fact which gives a new perspective on the automorphism group of Sn. [ABSTRACT FROM AUTHOR]

We construct automorphisms of positive entropy of K3 surfaces of Picard number 4 with certain Salem numbers. We also prove that there is a fixed point free automorphism of positive entropy on a K3 surface of Picard number 4 with Salem degree 4. [ABSTRACT FROM AUTHOR]

Consider a ring with an (anti)-automorphism ∇ of finite order. The fundamental aim of this manuscript is to introduce the notions of ∇-(semi)prime ideal and ∇-(semi)prime ring as a generalization of the notions of (semi)prime ideal, ∗ -(semi)prime ideal, (semi)prime ring and ∗ -(semi)prime ring. Furthermore, we will investigate their basic properties and study the commutative property of ∇-prime rings which satisfy certain central differential identities on ∇-ideals. [ABSTRACT FROM AUTHOR]

We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterization captures the phase transition of the symmetry of the random graphs. This phase transition coincides with that of the graph connectivity. We also show that there are unbounded degree sequences where these two thresholds do not coincide. [ABSTRACT FROM AUTHOR]