论文标题
带有$ M $ -DCI物业的二面体集团
Dihedral groups with the $m$-DCI property
论文作者
论文摘要
相对于子集的$ g $，相对于子集的$ g $的cayley digraph $ \ rm {cay}（g，s）$ $ g $的子集$ s $，如果有任何cayley digraph $ \ rm \ rm {cay}（cay}（g，g，t）$ iSomorphic to $ \ rm \ rm \ rm { \ rm {aut}（g）$，使得$ s^α= t $。对于正整数$ m $，如果所有Cayley Digraphs均为$ G $的$ M $ dci属性，则具有$ g $的$ M $ g $，均为$ m $ $ m $。 li [带有$ M $ -DCI物业的循环集团，欧洲J. Combin。 18（1997）655-665]以$ M $ -DCI属性为特征，在本文中，我们用$ M $ -DCI属性进行了二面体。对于二面体组$ \ mathrm {d} _ {2n} $ of订单$ 2N $，假设$ \ mathrm {d} _ {2n} $具有$ 1 \ leq m \ leq m \ leq n-1 $的$ m $ -dci属性。然后证明$ n $是奇怪的，如果$ p+1 \ leq m \ leq m \ leq n-1 $对于奇数prime divisor $ p $ $ n $，则是$ p^2 \ nmid n $。此外，如果$ n $是Prime $ Q $的功率，则$ \ Mathrm {d} _ {2n} $具有$ M $ -DCI属性，并且仅当$ n = q $或$ q $或$ q $是奇数且$ 1 \ leq m \ leq m \ leq q $。
A Cayley digraph $\rm{Cay}(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called a CI-digraph if for any Cayley digraph $\rm{Cay}(G,T)$ isomorphic to $\rm{Cay}(G,S)$, there is an $α\in \rm{Aut}(G)$ such that $S^α=T$. For a positive integer $m$, $G$ is said to have the $m$-DCI property if all Cayley digraphs of $G$ with out-valency $m$ are CI-digraphs. Li [The Cyclic groups with the $m$-DCI Property, European J. Combin. 18 (1997) 655-665] characterized cyclic groups with the $m$-DCI property, and in this paper, we characterize dihedral groups with the $m$-DCI property. For a dihedral group $\mathrm{D}_{2n}$ of order $2n$, assume that $\mathrm{D}_{2n}$ has the $m$-DCI property for some $1 \leq m\leq n-1$. Then it is shown that $n$ is odd, and if further $p+1\leq m\leq n-1$ for an odd prime divisor $p$ of $n$, then $p^2\nmid n$. Furthermore, if $n$ is a power of a prime $q$, then $\mathrm{D}_{2n}$ has the $m$-DCI property if and only if either $n=q$, or $q$ is odd and $1\leq m\leq q$.
