Orientably-Regular $p$-Maps and Regular $p$-Maps

Abstract

A map is called a {\it $p$-map} if it has a prime $p$-power vertices. An orientably-regular (resp. A regular ) $p$-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 the normal Sylow $p$-subgroup. In this paper, it will be proved that both orientably-regular $p$-maps and regular $p$-maps are solvable and except for few cases that $p\in \{2,3\}$, they are normal. Moreover, nonnormal $p$-maps will be characterized and some properties and constructions of normal $p$-maps will be given.