Automorphism Of Prime Order Research Articles

Nonsymplectic automorphisms of prime order on O’Grady’s sixfolds

We classify nonsymplectic automorphisms of prime order on irreducible holomorphic symplectic manifolds of O’Grady’s 6-dimensional deformation type. More precisely, we give a classification of the invariant and coinvariant sublattices of the second integral cohomology group.

On binary LCD codes possessing an automorphism of order 13

In this work we apply the method for constructing binary LCD codes via an automorphism of prime order described in [3] and [4]. Thus we obtain all optimal LCD codes of lengths 26, 27 and 28 possessing an automorphism of order 13 with two cycles.

Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

A right Engel sink of an element g of a group G is a set {{mathscr {R}}}(g) such that for every xin G all sufficiently long commutators [...[[g,x],x],dots ,x] belong to {mathscr {R}}(g). (Thus, g is a right Engel element precisely when we can choose {{mathscr {R}}}(g)={ 1}.) We prove that if a profinite group G admits a coprime automorphism varphi of prime order such that every fixed point of varphi has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set {{mathscr {E}}}(g) such that for every xin G all sufficiently long commutators [...[[x,g],g],dots ,g] belong to {{mathscr {E}}}(g). (Thus, g is a left Engel element precisely when we can choose {mathscr {E}}(g)={ 1}.) We prove that if a profinite group G admits a coprime automorphism varphi of prime order such that every fixed point of varphi has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

Exponent of a finite group admitting a coprime automorphism of prime order

Abstract Let 𝐺 be a finite group admitting an automorphism 𝜙 of prime order 𝑝 such that ( | G | , p ) = 1 (\lvert G\rvert,p)=1 . It is shown that if the fixed-point subgroup for 𝜙 has rank 𝑟 and ( x - 1 ⁢ x ϕ ) e = 1 (x^{-1}x^{\phi})^{e}=\nobreak 1 for each x ∈ G x\in G , then the exponent of [ G , ϕ ] [G,\phi] is ( e , p , r ) (e,p,r) -bounded.

On prime order automorphisms of generalized quadrangles

In this paper, we study prime order automorphisms of generalized quadrangles. We show that, if 𝒬 is a thick generalized quadrangle of order (s,t), where s>t and s+1 is prime, and 𝒬 has an automorphism of order s+1, then

Idempotent-fixing automorphisms of completely regular semigroups

The purpose of this note is to prove that if a completely regular semigroup with finite $$\mathscr {H}$$ -classes posesses an idempotent-fixing automorphism of prime order, then it is an over-nilpotent semigroup.

Complex ball quotients from manifolds of K3[n]-type

We describe periods of irreducible holomorphic symplectic manifolds of K3[n]-type with a non-symplectic automorphism of prime order p≥3. These turn out to lie on complex ball quotients and we are able to give a precise characterization of when the period map is bijective by introducing the notion of K(T)-generality.

有限生成剩余有限群的素数阶自同构

有限生成剩余有限群的素数阶自同构

On (q,n)-gonal pseudo-real Riemann surfaces

A compact Riemann surface [Formula: see text] of genus [Formula: see text] is called pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. In this paper, we study pseudo-real [Formula: see text]-gonal Riemann surfaces of genera greater or equal to two; these surfaces have anticonformal automorphisms of prime order [Formula: see text] such that the quotient spaces have genus [Formula: see text].

Prime order automorphisms of generalised Kummer fourfolds

We classify prime order automorphisms of generised Kummer fourfolds using lattice theory and recent results on ample cones and monodromy groups. We study various geometric realisations of the automorphisms obtained in the classification.

On automorphisms of a distance-regular graph with intersection array {39, 36, 1; 1, 2, 39}

Possible prime-order automorphisms and their fixed-point subgraphs are found for a hypothetical distance-regular graph with intersection array {39, 36, 1; 1, 2, 39}. It is shownthat graphs with intersection arrays {15, 12, 1; 1, 2, 15}, {35, 32, 1; 1, 2, 35}, and {39, 36, 1; 1, 2, 39} are not vertex-symmetric.

On prime degree isogenies between K3 surfaces

We classify prime order isogenies between algebraic K3 surfaces whose rational transcendental Hodges structures are not isometric. The morphisms of Hodge structures induced by these isogenies are correspondences by algebraic classes on the product fourfolds; however, they do not satisfy the hypothesis of the well-known Mukai--Nikulin theorem. As an application we describe isogenies obtained from K3 surfaces with an action of a symplectic automorphism of prime order.

Classification of automorphisms on a deformation family of hyper-Kähler four-folds by p -elementary lattices

We give a classification of all nonsymplectic automorphisms of prime order p acting on irreducible holomorphic symplectic four-folds deformation equivalent to the Hilbert scheme of two points on a K3 surface, for p = 2 , 3 , and 7 ≤ p ≤ 19 . Our classification relates some invariants of the fixed locus to the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients. In several cases we provide explicit examples. As an application, we find new examples of nonnatural nonsymplectic automorphisms of order 3 .

Induced automorphisms on irreducible symplectic manifolds:

We introduce the notion of induced automorphisms in order to state a criterion to determine whether a given automorphism on a manifold of K 3 [ n ] -type is, in fact, induced by an automorphism of a K 3 surface, and the manifold is a moduli space of stable objects on the K 3 . This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of K 3 [ 2 ] -type, and we prove that almost all cases are covered. Variations of this notion and the above criterion are introduced and discussed for the other known deformation types of irreducible symplectic manifolds. Furthermore, we provide a description of the picard lattice of several irreducible symplectic manifolds having a lagrangian fibration.

ON THE CONNECTEDNESS OF THE BRANCH LOCI OF NON-ORIENTABLE UNBORDERED KLEIN SURFACES OF LOW GENUS

AbstractThis paper is devoted to determine the connectedness of the branch loci of the moduli space of non-orientable unbordered Klein surfaces. We obtain a result similar to Nielsen's in order to determine topological conjugacy of automorphisms of prime order on such surfaces. Using this result we prove that the branch locus is connected for surfaces of topological genus 4 and 5.

Inverse semigroups with idempotent-fixing automorphisms

A celebrated result of J. Thompson says that if a finite group \(G\) has a fixed-point-free automorphism of prime order, then \(G\) is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.

Automorphisms of finite order of Nilpotent groups

Let $$\phi $$ be an automorphism of prime order $$p$$ of the nilpotent group $$G$$ . Frequently one needs to consider the map $$\psi : G\rightarrow G$$ given by $$\begin{aligned} g\psi =g\cdot g\phi \cdot g\phi ^2\cdot \ldots \cdot g\phi ^{p-1}. \end{aligned}$$ This $$\psi $$ is not usually a homomorphism of $$G$$ and $$ker\psi $$ and $$G\psi $$ need not be subgroups. Under relatively mild restrictions we prove that $$G\psi $$ is finite if and only if $$ker\psi $$ contains a subgroup of $$G$$ of finite index and $$ker\psi $$ is finite if and only if $$G\psi $$ contains a subgroup of finite index. These results do not extend to polycyclic groups or Cernikov groups.

On extremal self-dual codes of length 120

We prove that the only primes which may divide the order of the automorphism group of a putative binary self-dual doubly-even $$[120, 60, 24]$$ [ 120 , 60 , 24 ] code are $$2, 3, 5, 7, 19, 23$$ 2 , 3 , 5 , 7 , 19 , 23 and $$29$$ 29 . Furthermore we prove that automorphisms of prime order $$p \ge 5$$ p ? 5 have a unique cycle structure. Parts of the results are based on computer computations.

Self-orthogonal codes from some Bush-type Hadamard matrices

By means of a construction method outlined by Harada and Tonchev, we determine some non-binary self-orthogonal codes obtained from the row span of orbit matrices of Bush-type Hadamard matrices that admit a fixed-point-free and fixed-block-free automorphism of prime order. We show that the code [20, 15, 4]5 obtained from a (100, 45, 20) design is optimal, and those with parameters [36, 21, 6]3 and [20, 14, 4]5 obtained from a (36, 15, 6) and a (100, 45, 20) design respectively, are near-optimal for the given length and dimension. Furthermore, we obtained a conjecturally optimal self-dual doubly-even [72, 36, 12]2 code, and examined the code of an orbit matrix of a putative (676, 325, 156) design.

Groups Admitting an Automorphism of Prime Order with Finite Centralizer

We consider a group G with an automorphism of finite, usually prime, order. If G has finite Hirsch number, and also if G satisfies various stronger rank restrictions, we study the consequences and equivalent hypotheses of having only finitely many fixed-points. In particular we prove that if a group G with finite Hirsch number $${\mathfrak{h}}$$ admits an automorphism $${\varphi}$$ of prime order p such that $${\vert C_{G}(\varphi) \vert = n < \infty,}$$ then G has a subgroup of finite index bounded in terms of p, n and $${\mathfrak{h}}$$ that is nilpotent of p-bounded class.