arXiv subjects

Naer Wang

Publications and source records attributed to Naer Wang.

4 recordsLinked to original sources

arXiv2018

Smooth skew-morphisms of the dihedral groups

A skew-morphism $φ$ of a finite group $A$ is a permutation on $A$ such that $φ(1)=1$ and $φ(xy)=φ(x)φ^{π(x)}(y)$ for all $x,y\in A$ where $π:A\to\mathbb{Z}_{|φ|}$ is an integer function. A skew-morphism is smooth if $π(φ(x))=π(x)$ for all $x\in A$. The concept of smooth skew-morphisms is a generalization of that of $t$-balanced skew-morphisms. The aim of the paper is to develop a general theory of smooth skew-morphisms. As an application we classify smooth skew-morphisms of the dihedral groups.

math.GR↗

arXiv2018

Complete regular dessins of odd prime power order

A dessin is a $2$-cell embedding of a connected $2$-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts regularly on the edges. In this paper we employ group-theoretic method to determine and enumerate the isomorphism classes of regular dessins with the complete bipartite underlying graphs of odd prime power order.

math.CO↗

arXiv2018

Nilpotent groups of class two which underly a unique regular dessin

A dessin is an embedding of connected bipartite graph into an oriented closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts transitively on the edges. In the present paper regular dessins with a nilpotent automorphism group are investigated, and attention are paid on those with the highest level of external symmetry. Depending on the algebraic theory of dessins and using group-theoretical methods, we present a classification of nilpotent groups of class two which underly a unique regular dessin.

math.GR↗

arXiv2018

Regular dessins uniquely determined by a nilpotent automorphism group

It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is reduced to the classification of finite maximally automorphic $p$-groups $G$, i.e., the order of the automorphism group of $G$ attains Hall's upper bound. Maximally automorphic $p$-groups of nilpotency class three are classified.

math.GR↗