The Cyclic Groups with them-DCI Property

AbstractFor a finite groupGand a subsetSofGwhich does not contain the identity ofG,let Cay(G,S)denote the Cayley graph ofGwith respect toS.If, for all subsetsS, TofGof sizem,Cay(G,S)≅Cay(G,T)impliesSα=Tfor someα∈Aut(G), thenGis said to have them-DCI property. In this paper, a classification is presented of the cyclic groups with them-DCI property, which is reasonably complete

On numerically pluricanonical cyclic coverings

In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of surfaces of general type with p_g=0 and Miyaoka--Yau surfaces; in particular, they provide new examples of multicomponent moduli spaces of surfaces with given Chern numbers as well as new examples of surfaces that are not deformation equivalent to their complex conjugates.Comment: Revised version, 21 pages, submitted to Izvestiya: Mathematics. Some results in Section 3 are improved (in particular, better lower bounds in new Corollary 4 than in former Corollary 2). A few mirsprints are corrected. Several references are adde

Generalized quaternion groups with the mm-DCI property

A Cayley digraph Cay(G,S) of a finite group $G$ with respect to a subset $S$ of $G$ is said to be a CI-digraph if for every Cayley digraph Cay(G,T) isomorphic to Cay(G,S), there exists an automorphism $\sigma$ of $G$ such that $S^\sigma=T$. A finite group $G$ is said to have the $m$-DCI property for some positive integer $m$ if all $m$-valent Cayley digraphs of $G$ are CI-digraphs, and is said to be a DCI-group if $G$ has the $m$-DCI property for all $1\leq m\leq |G|$. Let $\mathrm{Q}_{4n}$ be a generalized quaternion group of order $4n$ with an integer $n\geq 3$, and let $\mathrm{Q}_{4n}$ have the $m$-DCI property for some $1 \leq m\leq 2n-1$. It is shown in this paper that $n$ is odd, and $n$ is not divisible by $p^2$ for any prime $p\leq m-1$. Furthermore, if $n\geq 3$ is a power of a prime $p$, then $\mathrm{Q}_{4n}$ has the $m$-DCI property if and only if $p$ is odd, and either $n=p$ or $1\leq m\leq p$.Comment: 1