Orientably-Regular $\pi$-Maps and Regular $\pi$-Maps

Abstract

Given a map with underlying graph $\mathcal{G}$, if the set of prime divisors of $|V(\mathcal{G}|$ is denoted by $\pi$, then we call the map a {\it $\pi$-map}. An orientably-regular (resp. A regular ) $\pi$-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 $\pi$-Hall subgroup. In this paper, it will be proved that orientably-regular $\pi$-maps are solvable and normal if $2\notin \pi$ and regular $\pi$-maps are solvable if $2\notin \pi$ and $G$ has no sections isomorphic to ${\rm PSL}(2,q)$ for some prime power $q$. In particular, it's shown that a regular $\pi$-map with $2\notin \pi$ is normal if and only if $G/O_{2^{'}}(G)$ is isomorphic to a Sylow $2$-group of $G$. Moreover, nonnormal $\pi$-maps will be characterized and some properties and constructions of normal $\pi$-maps will be given in respective sections.