Free Automorphisms Research Articles

Automorphism groups of quandles arising from groups

Let $G$ be a group and $\varphi \in \Aut(G)$. Then the set $G$ equipped with the binary operation $a*b=\varphi(ab^{-1})b$ gives a quandle structure on $G$, denoted by $\Alex(G, \varphi)$ and called the generalised Alexander quandle. When $G$ is additive abelian and $\varphi = -\id_G$, then $\Alex(G, \varphi)$ is the well-known Takasaki quandle. In this paper, we determine the group of automorphisms and inner automorphisms of Takasaki quandles of abelian groups with no 2-torsion, and Alexander quandles of finite abelian groups with respect to fixed-point free automorphisms. As an application, we prove that if $G\cong (\mathbb{Z}/p \mathbb{Z})^n$ and $\varphi$ is multiplication by a non-trivial unit of $\mathbb{Z}/p \mathbb{Z}$, then $\Aut\big(\Alex(G, \varphi)\big)$ acts doubly transitively on $\Alex(G, \varphi)$. This generalises a recent result of \cite{Ferman} for quandles of prime order.

Some Remarks on Commuting Fixed Point Free Automorphisms of Groups

In this a‌r‌t‌i‌c‌l‌e we will find n‌e‌c‌e‌s‌s‌a‌r‌y and s‌u‌f‌f‌i‌c‌i‌e‌n‌t c‌o‌n‌d‌i‌t‌i‌o‌n‌s for a f‌i‌x‌e‌d p‌o‌i‌n‌t f‌r‌e‌e autom‌o‌r‌p‌h‌i‌s‌m (f‌p‌f a‌u‌t‌o‌m‌o‌r‌p‌h‌i‌s‌m) o‌f a g‌r‌o‌u‌p t‌o b‌e a c‌o‌m‌m‌u‌t‌i‌n‌g a‌u‌t‌o‌m‌o‌r‌p‌h‌i‌s‌m. F‌o‌r a given prim‌e ‌ we f‌i‌n‌d t‌h‌e s‌m‌a‌l‌l‌e‌s‌t o‌r‌d‌e‌r o‌f a n‌o‌n a‌b‌e‌l‌i‌a‌n p-g‌r‌o‌u‌p a‌d‌m‌i‌t‌t‌i‌n‌g a c‌o‌m‌m‌u‌t‌i‌n‌g f‌i‌x‌e‌d p‌o‌i‌n‌t f‌r‌e‌e a‌u‌t‌o‌m‌o‌r‌p‌h‌i‌s‌m. W‌e p‌r‌o‌v‌e t‌h‌a‌t a g‌r‌o‌u‌p o‌f o‌r‌d‌e‌r p3 h‌a‌v‌i‌n‌g a c‌o‌m‌m‌u‌t‌i‌n‌g f‌p‌f autom‌o‌r‌p‌h‌i‌s‌m, h‌a‌s a r‌e‌s‌t‌r‌i‌c‌t‌e‌d s‌t‌r‌u‌c‌t‌u‌r‌e. M‌o‌r‌e‌o‌v‌e‌r w‌e p‌r‌o‌v‌e t‌h‌a‌t i‌f a f‌i‌n‌i‌t‌e g‌r‌o‌u‌p a‌d‌m‌i‌t‌s a f‌p‌f au‌t‌o‌m‌o‌r‌p‌h‌i‌s‌m o‌f o‌r‌d‌e‌r 4, t‌h‌e‌n t‌h‌e c‌o‌n‌v‌e‌r‌s‌e of Laffey’s result holds in G1.

On involutions in extremal self-dual codes and the dual distance of semi self-dual codes

A classical result of Conway and Pless is that a natural projection of the fixed code of an automorphism of odd prime order of a self-dual binary linear code is self-dual [13]. In this paper we prove that the same holds for involutions under some (quite strong) conditions on the codes.In order to prove it, we introduce a new family of binary codes: the semi self-dual codes. A binary self-orthogonal code is called semi self-dual if it contains the all-ones vector and is of codimension 2 in its dual code. We prove upper bounds on the dual distance of semi self-dual codes.As an application we get the following: let C be an extremal self-dual binary linear code of length 24m and σ∈Aut(C) be a fixed point free automorphism of order 2. If m is odd or if m=2k with (5k−1k−1) odd then C is a free F2〈σ〉-module. This result has quite strong consequences on the structure of the automorphism group of such codes.

REVISIT TO CONNECTED ALEXANDER QUANDLES OF SMALL ORDERS VIA FIXED POINT FREE AUTOMORPHISMS OF FINITE ABELIAN GROUPS

In this paper we provide a rigorous proof for the fact that there are exactly 8 connected Alexander quandles of order <TEX>$2^5$</TEX> by combining properties of fixed point free automorphisms of finite abelian 2-groups and the classification of conjugacy classes of GL(5, 2). Furthermore we verify that six of the eight associated Alexander modules are simple, whereas the other two are semisimple.

The Cayley-Oguiso automorphism of positive entropy on a K3 surface

Recently Oguiso showed the existence of K3 surfaces that admit a fixedpoint free automorphism of positive entropy. The K3 surfaces used byOguiso have a particular rank two Picard lattice. We show, usingresults of Beauville, that these surfaces are therefore determinantalquartic surfaces. Long ago, Cayley constructed an automorphism ofsuch determinantal surfaces. We show that Cayley's automorphismcoincides with Oguiso's free automorphism. We also exhibit anexplicit example of a determinantal quartic whose Picard lattice hasexactly rank two and for which we thus have an explicit description ofthe automorphism.

Fixed point free automorphisms of groups related to finite fields

Let G=Fq⋊〈β〉 be the semidirect product of the additive group of the field of q=pn elements and the cyclic group of order d generated by the invertible linear transformation β defined by multiplication by a power of a primitive root of Fq. We find an arithmetic condition on d so that every endomorphism of G is determined by its values on (1,1) and (0,β). When that is the case, we determine the fixed point free automorphisms of G. If d equals the odd part of q−1 then we count the fixed point free automorphisms of G—such exist only when p is a Fermat prime.

On the automorphism group of inversive planes of odd order

The structure of the fix bundle free automorphism groups of inversive planes of odd order is determined. As a special case of our main result, the automorphism groups with a transitive action on the points of an inversive plane of odd order are essentially determined, and the plane is shown to be miquelian when these have no non-trivial normal subgroups of odd order.

Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties

We define Enriques varieties as a higher dimensional generalization of Enriques surfaces and construct examples by using fixed point free automorphisms on generalized Kummer varieties. We also classify all automorphisms of generalized Kummer varieties that come from an automorphism of the underlying abelian surface.

Hermitian modular forms congruent to 1 modulo p

For any natural number $$\ell$$ and any prime $$p \equiv 1$$ (mod 4) not dividing $$\ell$$ there is a Hermitian modular form of arbitrary genus n over $$L := {\mathbb{Q}}[\sqrt{-\ell}]$$ that is congruent to 1 modulo p which is a Hermitian theta series of an O L -lattice of rank p − 1 admitting a fixed point free automorphism of order p. It is shown that also for non-free lattices such theta series are modular forms.

On Existence of Semifields with Free Automorphism Groups

The aim of this paper is to prove the existence of a semifield A of order q4 where q is a power of 2, having the properties, A contains GF(q2) in its left-nucleus and admits a free four-group E ≅ Z2 × Z2 of automorphisms.

Linear orders with distinguished function symbol

We consider certain linear orders with a function on them, and discuss for which types of functions the resulting structure is or is not computably categorical. Particularly, we consider computable copies of the rationals with a fixed-point free automorphism, and also ω with a non-decreasing function.

Semifields with free automorphism groups

The purpose of this article is to extend the results in [AlAli M.; Hering C.; Neumann A. and Rawashdeh A.: On the existence of semifields of prime power order admitting free automorphism groups, to appear in J. Geom.] and to describe all semifields of order q 4 , q is an odd prime power greater than 3, admitting free automorphism groups isomorphic to Z 2 × Z 2 . 2000 Mathematics Subject Classification: 12K10, 51A40.

A Semifield of Order 54 Admitting a Free 4-Group of Automorphisms

The purpose of this article is to give classification of semifields of order 54 admitting a free automorphism group E≅ z2 × z2, and to find their kernels (left nuclei).

Forbidden retracts for finite ordered sets of width at most four

The relationship between the fixed point property and forbidden retracts associated with a forgetful functor is formulated. Finite ordered sets of width at most four with fixed point free automorphisms are described. Linear time algorithms for deciding whether a finite ordered set of width two has the fixed point property and whether a finite ordered set of width at most three has a fixed point free automorphism are presented.

On the existence of semifields of prime power order admitting free automorphism groups

The purpose of this paper is to prove the existence of semifields of order q4 for any odd prime power q = p r , q > 3, admitting a free automorphism group isomorphic to Z2 × Z2.

Formula omitted]-graded Lie rings

Thompson proved that every finite group with a fixed point free automorphism of prime order is nilpotent and G. Higman proved that the nilpotency class is bounded in terms of the prime alone. Kreknin and Kostrikin produced the first explicit bound by reducing to the problem of bounding the nilpotency class of a Z p -graded Lie ring L with L 0 = 0 . Meixner later improved this bound. A step in the proof of Kreknin and Kostrikin is to bound the derived length of Z n -graded Lie rings L with L 0 = 0 . In this paper we improve these bounds.

Finite groups admitting fixed-point free automorphisms of order pqr

Finite groups admitting fixed-point free automorphisms of order pqr

Cubic self-dual binary codes

We study binary self-dual codes with a fixed point free automorphism of order three. All binary codes of that type can be obtained by a cubic construction that generalizes Turyn's. We regard such "cubic" codes of length 3/spl lscr/ as codes of length /spl lscr/ over the ring F/sub 2//spl times/F/sub 4/. Classical notions of Type II codes, shadow codes, and weight enumerators are adapted to that ring. Two infinite families of cubic codes are introduced. New extremal binary codes in lengths /spl les/ 66 are constructed by a randomized algorithm. Necessary conditions for the existence of a cubic [72,36,16] Type II code are derived.

ℤ3-Twisted Representations of Lattice Vertex Operator Algebras

Let L be a positive definite even lattice and let g ∈ Aut L be a fixed point free automorphism of order 3. We determine the twisted Zhu's algebra A ĝ (V L ) for the lattice vertex operator algebra V L , where ĝ is an automorphism of V L induced from g. As a result, we show that the set of all irreducible ĝ-twisted modules for V L (up to isomorphism) are exactly those constructed by Dong and Lepowsky (1996) and Lepowsky (1985).

K-loops derived from Frobenius groups

We consider a generalization of the representation of the so-called co-Minkowski plane (due to H. and R. Struve) to an abelian group ( V,+) and a commutative subgroup G of Aut( V,+). If P= G× V satisfies suitable conditions then an invariant reflection structure (in the sense of Karzel (Discrete Math. 208/209 (1999) 387–409)) can be introduced in P which carries the algebraic structure of K-loop on P (cf. Theorem 1). We investigate the properties of the K-loop ( P,+) and its connection with the semi-direct product of V and G. If G is a fixed point free automorphism group then it is possible to introduce in ( P,+) an incidence bundle in such a way that the K-loop ( P,+) becomes an incidence fibered loop (in the sense of Zizioli (J. Geom. 30 (1987) 144–151)) (cf. Theorem 3).