论文标题
有效的规范$π$ - 图和常规$π$ - 图
Orientably-Regular $π$-Maps and Regular $π$-Maps
论文作者
论文摘要
如果$ | v（\ Mathcal {g} | $表示$π$，则给定带有基础图$ \ MATHCAL {g} $的地图，如果$ | 如果所有定向保护自动形态的组$ g^+$ groveriable（ress.g^+$）是有效的（resp。一个常规）$π$ -MAP称为{\ IT可溶解}（sup。$ g^+$ $ g^+$ $ g^+$（分别是自动形态学的组$ g $）；如果$ g^+$（resp。$ g $）包含一个普通的$π$ -hall子组，则称为{\ it normal}。 在本文中，可以证明，如果$ 2 \notinπ$ $ 2 \notinπ$，并且常规$π$ -maps如果$ 2 \ notinπ$和$ g $没有各个部分isomorphic isomorphic to $ {\ rmmpsl}（2，Q）$ q for Some Q）。特别是，当$ g/o_ {2^{'}}（g）$仅当$ g/o_ g/o_ {2^{'}}（g）$与$ g $的Sylow $ 2 $ -GROUP是同构时，它的常规$π$ - 图是正常的，并且仅当$ g/o_ {2^{'}}（g）是同构的。 此外，将表征非正常的$π$图，并在各个部分中给出正常$π$ - 图的某些属性和构造。
Given a map with underlying graph $\mathcal{G}$, if the set of prime divisors of $|V(\mathcal{G}|$ is denoted by $π$, then we call the map a {\it $π$-map}. An orientably-regular (resp. A regular ) $π$-map is called {\it solvable} if the group $G^+$ of all orientation-preserving automorphisms (resp. the group $G$ of automorphisms) is solvable; and called {\it normal} if $G^+$ (resp. $G$) contains a normal $π$-Hall subgroup. In this paper, it will be proved that orientably-regular $π$-maps are solvable and normal if $2\notin π$ and regular $π$-maps are solvable if $2\notin π$ and $G$ has no sections isomorphic to ${\rm PSL}(2,q)$ for some prime power $q$. In particular, it's shown that a regular $π$-map with $2\notin π$ is normal if and only if $G/O_{2^{'}}(G)$ is isomorphic to a Sylow $2$-group of $G$. Moreover, nonnormal $π$-maps will be characterized and some properties and constructions of normal $π$-maps will be given in respective sections.
