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

TL;DR

This paper investigates the properties of orientably-regular and regular $ ext{ extpi}$-maps, establishing conditions for their solvability and normality based on the prime divisors of the underlying graph's vertices.

Contribution

It proves that orientably-regular $ ext{ extpi}$-maps are solvable and normal if 2 is not in $ ext{ extpi}$, and characterizes when regular $ ext{ extpi}$-maps are normal, expanding understanding of their algebraic structure.

Findings

Orientably-regular $ ext{ extpi}$-maps are solvable and normal if 2 not in $ ext{ extpi}$.

Regular $ ext{ extpi}$-maps are solvable if 2 not in $ ext{ extpi}$ and $G$ has no sections isomorphic to ${ m PSL}(2,q)$.

A regular $ ext{ extpi}$-map with 2 not in $ ext{ extpi}$ is normal iff $G/O_{2^{'}}(G)$ is a Sylow 2-group.

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.