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

TL;DR

This paper investigates the structure of $p$-maps, showing that most are solvable and normal except for specific small primes, and characterizes nonnormal cases with properties and constructions.

Contribution

It proves that orientably-regular and regular $p$-maps are mostly solvable and normal, providing classifications and constructions for nonnormal $p$-maps.

Findings

Most $p$-maps are solvable and normal for primes p > 3.

Nonnormal $p$-maps are characterized and constructed.

Exceptions occur for primes p in {2, 3}.

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.