2022/09/16 by Li, Xiaogang, Tian, Yao
#05C65 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2209.07991
Given a map with underlying graph G, if the set of prime divisors of |V(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∉ π and regular π-maps are solvable if 2∉ 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∉ π is normal if and only if G/O2'(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.