Given a map with underlying graph G, if the set of prime divisors
of β£V(Gβ£ is denoted by Ο, then we call the map a {\it
Ο-map}.
An orientably-regular (resp. A regular ) Ο-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 Ο-Hall subgroup.
In this paper, it will be proved that orientably-regular Ο-maps are
solvable and normal if 2β/Ο and regular Ο-maps are solvable if
2β/Ο and G has no sections isomorphic to PSL(2,q) for some
prime power q. In particular, it's shown that a regular Ο-map with
2β/Ο is normal if and only if G/O2β²β(G) is isomorphic to a
Sylow 2-group of G.
Moreover, nonnormal Ο-maps will be characterized and some properties and
constructions of normal Ο-maps will be given in respective sections.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:2201.0430