Automorphism Research Articles

English

Consider T n (F)—the ring of all n × n upper triangular matrices defined over some field F. A map φ is called a zero product preserver on T n (F) in both directions if for all x, y ∈ T n (F) the condition xy = 0 is satisfied if and only if φ(x)φ(y) = 0. In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map φ may act in any bijective way, whereas for the zero divisors and zero matrix one can write φ as a composition of three types of maps. The first of them is a conjugacy, the second one is an automorphism induced by some field automorphism, and the third one transforms every matrix x into a matrix x′ such that {y ∈ T n (F): xy = 0} = {y ∈ T n (F): x′y = 0}, {y ∈ T n (F): yx = 0} = {y ∈ T n (F): yx′ = 0}.

Comments on the Golden Partition Conjecture

We generalize the result of Zaguia that 1/3--2/3 Conjecture is satisfied by every N-free finite poset which is not a chain: we show a wider class of posets which satisfy the Golden Partition Conjecture. We generalize the result of Pouzet that 1/3--2/3 Conjecture is satisfied by every finite poset with a non-trivial automorphism: we show that such posets satisfy the Golden Partition Conjecture.

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model mathcal {M} of set theory, and a nontrivial automorphism j of mathcal {M}, let mathcal {I}_{mathrm {fix}}(j) be the submodel of mathcal {M} whose universe consists of elements m of mathcal {M} such that j(x)=x for every x in the transitive closure of m (where the transitive closure of m is computed within mathcal {M}). Here we study the class mathcal {C} of structures of the form mathcal {I}_{mathrm {fix}}(j), where the ambient model mathcal {M} satisfies a frugal yet robust fragment of mathrm {ZFC} known as mathrm {MOST}, and j(m)=m whenever m is a finite ordinal in the sense of mathcal {M}. Our main achievement is the calculation of the theory of mathcal {C} as precisely mathrm {MOST+Delta }_{0}^{mathcal {P}}-mathrm {Collection}. The following theorems encapsulate our principal results:Theorem A.Every structure inmathcal {C}satisfiesmathrm {MOST+Delta }_{0}^{mathcal {P}}-mathrm { Collection}.Theorem B. Each of the following three conditions is sufficient for a countable structuremathcal {N}to be in mathcal {C}:(a) mathcal {N}is a transitive model ofmathrm {MOST+Delta }_{0}^{mathcal {P}}-mathrm {Collection}.(b) mathcal {N}is a recursively saturated model ofmathrm {MOST+Delta }_{0}^{mathcal {P}}-mathrm {Collection}.(c) mathcal {N}is a model of mathrm {ZFC}.Theorem C. Suppose mathcal {M} is a countable recursively saturated model of mathrm {ZFC} and I is a proper initial segment of mathrm {Ord}^{mathcal {M}} that is closed under exponentiation and containsomega ^mathcal {M}. There is a group embeddingjlongmapsto check{j}from mathrm {Aut}(mathbb {Q}) into mathrm {Aut}(mathcal {M}) such that Iis the longest initial segment ofmathrm {Ord}^{mathcal {M}}that is pointwise fixed bycheck{j}for every nontrivial jin mathrm {Aut}(mathbb {Q}).In Theorem C, mathrm {Aut}(X) is the group of automorphisms of the structure X, and mathbb {Q} is the ordered set of rationals.

Groups of automorphisms of local fields of period p and nilpotent class < p, II

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be the maximal quotient of period [Formula: see text] and nilpotent class [Formula: see text] of the Galois group of a maximal [Formula: see text]-extension of [Formula: see text]. We describe the ramification filtration [Formula: see text] and relate it to an explicit form of the Demushkin relation for [Formula: see text]. The results are given in terms of Lie algebras attached to the appropriate [Formula: see text]-groups by the classical equivalence of the categories of [Formula: see text]-groups and Lie algebras of nilpotent class [Formula: see text].

Bounds for Distinguishing Invariants of Infinite Graphs

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.

Distinguishing chromatic number of random Cayley graphs

The distinguishing chromatic number of a graph G, denoted χD(G), is defined as the minimum number of colors needed to properly color G such that no non-trivial automorphism of G fixes each color class of G. In this paper, we consider random Cayley graphs Γ defined over certain abelian groups A with |A|=n, and show that with probability at least 1−n−Ω(logn), χD(Γ)≤χ(Γ)+1.

Automorphisms of pure braid groups

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to an automorphism of $B_n$.

Tautological rings on Jacobian varieties of curves with automorphisms

Let $J$ be the Jacobian of a smooth projective complex curve $C$ which admits non-trivial automorphisms, and let $A(J)$ be the ring of algebraic cycles on $J$ with rational coefficients modulo algebraic equivalence. We present new tautological rings in $A(J)$ which extend in a natural way the tautological ring studied by Beauville (Compos Math 140(3):683-688, 2004). We then show there exist tautological rings induced on special complementary abelian subvarieties of $J$.

Breaking graph symmetries by edge colourings

The distinguishing index D′(G) of a graph G is the least number of colours needed in an edge colouring which is not preserved by any non-trivial automorphism. Broere and Pilśniak conjectured that if every non-trivial automorphism of a countable graph G moves infinitely many edges, then D′(G)≤2. We prove this conjecture.

Groups of automorphisms of local fields of period p and nilpotent class < p, I

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be a maximal [Formula: see text]-extension of [Formula: see text] with the Galois group of period [Formula: see text] and nilpotent class [Formula: see text]. In this paper, we develop formalism which allows us to study the structure of [Formula: see text] via methods of Lie theory. In particular, we introduce an explicit construction of a Lie [Formula: see text]-algebra [Formula: see text] and an identification [Formula: see text], where [Formula: see text] is a [Formula: see text]-group obtained from the elements of [Formula: see text] via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration [Formula: see text] and an explicit form of the Demushkin relation for [Formula: see text].

Game distinguishing numbers of Cartesian products

The distinguishing number of a graph H is a symmetry related graph invariant whose study started two decades ago. The distinguishing number D ( H ) is the least integer d such that H has a distinguishing d -coloring. A distinguishing d -coloring is a coloring c : V ( H ) → {1, …, d } invariant only under the trivial automorphism. In this paper, we continue the study of a game variant of this parameter, recently introduced. The distinguishing game is a game with two players, Gentle and Rascal, with antagonistic goals. This game is played on a graph H with a fixed set of d ∈ N colors. Alternately, the two players choose a vertex of H and color it with one of the d colors. The game ends when all the vertices have been colored. Then Gentle wins if the d -coloring is distinguishing and Rascal wins otherwise. This game defines two new invariants, which are the minimum numbers of colors needed to ensure that Gentle has a winning strategy, depending who starts the game. The invariant could be infinite. In this paper, we focus on the Cartesian product, a graph operation well studied in the classical case. We give sufficient conditions on the order of two connected factors H and F that are relatively prime, which ensure that one of the game distinguishing numbers of the Cartesian product H ▫ F is finite. If H is a so-called involutive graph, we give an upper bound of order D 2 ( H ) for one of the game distinguishing numbers of H ▫ F . Finally, using in part the previous result, we compute the exact value of these invariants for Cartesian products of relatively prime cycles. It turns out that the value is either infinite or equal to 2 , depending on the parity of the product order.

Edge motion and the distinguishing index

The distinguishing indexD′(G) of a graph G is the least number d such that G has an edge colouring with d colours that is preserved only by the trivial automorphism. We investigate the edge motion of a graph with respect to its automorphisms and compare it with the vertex motion. We prove an analog of the Motion Lemma of Russell and Sundaram, and we use it to determine the distinguishing index of powers of complete graphs and of cycles with respect to the Cartesian, direct and strong product.

Families of exotic affine 3-spheres

We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non-algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non-isomorphic exotic affine 3-spheres parametrized by the closed points of S.

A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS

AbstractMilnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$d the singular locus of Md and by ${\mathcal B}$d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$2 is a cubic curve; so ${\mathcal B}$2 is connected and ${\mathcal S}$2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$d = ${\mathcal B}$d. In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$d.

Improving upper bounds for the distinguishing index

The distinguishing index of a graph G , denoted by D ʹ( G ) , is the least number of colours in an edge colouring of G not preserved by any non-trivial automorphism. We characterize all connected graphs G with D ʹ( G ) ≥ Δ ( G ) . We show that D ʹ( G ) ≤ 2 if G is a traceable graph of order at least seven, and D ʹ( G ) ≤ 3 if G is either claw-free or 3 -connected and planar. We also investigate the Nordhaus-Gaddum type relation: 2 ≤ D ʹ( G ) + D ʹ( ‾ G ) ≤ max{Δ ( G ), Δ ( ‾ G )} + 2 and we confirm it for some classes of graphs.

Improved bounds for hypohamiltonian graphs

A graph $G$ is hypohamiltonian if $G$ is non-hamiltonian and $G - v$ is hamiltonian for every $v \in V(G)$. In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at most 17. In this article, we present an algorithm to generate all graphs of a given order and apply it to prove that there exist exactly 14 graphs of order 18 and 34 graphs of order 19. We also extend their results in the cubic case. Furthermore, we show that (i) the smallest graph of girth 6 has order 25, (ii) the smallest planar graph has order at least 23, (iii) the smallest cubic planar graph has order at least 54, and (iv) the smallest cubic planar graph of girth 5 with non-trivial automorphism group has order 78.

Fano plane’s embeddings on compact orientable surfaces

In this paper we study embeddings of the Fano plane as a bipartite graph. We classify all possible embeddings especially focusing on those with non-trivial automorphism group. We study them in terms of rotation systems, isomorphism classes and chirality. We construct quotients and show how to obtain information about face structure and genus of the covering embedding. As a by-product of the classification we determine the genus polynomial of the Fano plane.

Quantum graphs: PT-symmetry and reflection symmetry of the spectrum

Not necessarily self-adjoint quantum graphs—differential operators on metric graphs—are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) P. If the differential operator is PT-symmetric, then its spectrum has reflection symmetry with respect to the real line. Our goal is to understand whether the opposite statement holds, namely, whether the reflection symmetry of the spectrum of a quantum graph implies that the underlying metric graph possesses a non-trivial automorphism and the differential operator is PT-symmetric. We give partial answer to this question by considering equilateral star-graphs. The corresponding Laplace operator with Robin vertex conditions possesses reflection-symmetric spectrum if and only if the operator is PT-symmetric with P being an automorphism of the metric graph.

Automorphisms of Toeplitz $\mathscr {B}$-free systems

Automorphisms of Toeplitz $\mathscr {B}$-free systems

Distinguishing number and distinguishing index of certain graphs

The distinguishing number (index) D(G) (D0(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. In this paper we compute these two parameters for some specific graphs. Also we study the distinguishing number and the distinguishing index of corona product of two graphs.