The automorphism group and fixing number of orthogonality graph over a vector space

Let [Formula: see text] be a field, and [Formula: see text] the [Formula: see text]-dimensional row vector space over [Formula: see text]. The orthogonality graph [Formula: see text] of [Formula: see text] is an undirected simple graph which has [Formula: see text] as its vertex set, and for distinct [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the transpose of [Formula: see text]. When [Formula: see text] is finite, it is shown that any automorphism of [Formula: see text] can be decomposed into the product of a row-orthogonal automorphism and either a permutation automorphism or a field automorphism; moreover, the fixing number and metric dimension of [Formula: see text] are considered.

Continuous automorphisms of Cremona groups

We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.

Recognition of symplectic and orthogonal groups of small dimensions by spectrum

We refer to the set of the orders of elements of a finite group as its spectrum and say that finite groups are isospectral if their spectra coincide. In this paper, we determine all finite groups isospectral to the simple groups [Formula: see text], [Formula: see text], and [Formula: see text]. In particular, we prove that with just four exceptions, every such finite group is an extension of the initial simple group by a (possibly trivial) field automorphism.

Reidemeister spectrum of special and general linear groups over some fields contains 1

We prove that if [Formula: see text] is an algebraically closed field of zero characteristic which has infinite transcendence degree over [Formula: see text], then there exists a field automorphism [Formula: see text] of [Formula: see text] and [Formula: see text] such that [Formula: see text]. This fact implies that [Formula: see text] and [Formula: see text] do not possess the [Formula: see text]-property. However, if the transcendece degree of [Formula: see text] over [Formula: see text] is finite, then [Formula: see text] and [Formula: see text] are known to possess the [Formula: see text]-property [13].

A Note on Infinite Type Germs of a Real Hypersurface in

The purpose of this article is to show that there exists a smooth real hypersurface germ of D'Angelo infinite type in such that it does not admit any (singular) holomorphic curve that has infinite order contact with at . 2010 Mathematics Subject Classification. Primary 32T25; Secondary 32C25. Key words and phrases: Holomorphic vector field, automorphism group, real hypersurface, infinite type point.

Automorphisms of the co-maximal ideal graph over matrix ring

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] be the ring of all [Formula: see text] matrices over [Formula: see text], [Formula: see text] be the set of all nontrivial left ideals of [Formula: see text]. The co-maximal ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph with [Formula: see text] as vertex set and two nontrivial left ideals [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text]. If [Formula: see text], it is easy to see that [Formula: see text] is a complete graph, thus any permutation of vertices of [Formula: see text] is an automorphism of [Formula: see text]. A natural problem is: How about the automorphisms of [Formula: see text] when [Formula: see text]. In this paper, we aim to solve this problem. When [Formula: see text], a mapping [Formula: see text] on [Formula: see text] is proved to be an automorphism of [Formula: see text] if and only if there is an invertible matrix [Formula: see text] and a field automorphism [Formula: see text] of [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text].

Automorphisms of a commuting graph of rank one upper triangular matrices

Let $F$ be a finite field, $n\geqslant 2$ an arbitrary integer, $\mathcal{M}_n(F)$ the set of all $n\times n$ matrices over $F$, and $\mathcal{U}_n^1(F)$ the set of all rank one upper triangular matrices of order $n$. For $\mathcal{S}\subseteq\mathcal{M}_n(F)$, denote $C(\mathcal{S})=\{X\in \mathcal{S} |\ XA=AX \ \hbox{for all}\ A\in \mathcal{S}\}$. The commuting graph of $\mathcal{S}$, denoted by $\Gamma(\mathcal{S})$, is the simple undirected graph with vertex set $\mathcal{S}\setminus C(\mathcal{S})$ in which for every two distinct vertices $A$ and $B$, $A\sim B$ is an edge if and only if $AB=BA$. In this paper, it is shown that any graph automorphism of $\Gamma(\mathcal{U}_n^1(F))$ with $n\geqslant 3$ can be decomposed into the product of an extremal automorphism, an inner automorphism, a field automorphism and a local scalar multiplication.

Representations by Hermitian Forms in a Finite Field of Characteristic Two

Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.