A map is called a p-map if it has a prime p-power vertices. An orientably-regular (resp. A regular) p-map is called solvable if the group G+ of all orientation-preserving automorphisms (resp. the ...group G of automorphisms) is solvable; and called 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∈{2,3}, they are normal. Moreover, nonnormal p-maps will be characterized and some properties and constructions of normal p-maps will be given.
Given an orientable map
M
, and an integer
e
relatively prime to the valency of
M
, the
e
th rotational power
M
e
of
M
is the map formed by replacing the cyclic rotation of edges around each vertex ...with its
e
th power. If
M
and
M
e
are isomorphic, and the corresponding isomorphism preserves the orientation of the carrier surface, then we say that
e
is an exponent of
M
.In this paper, we use canonical regular covers of maps to prove that for every given hyperbolic pair (
k
,
m
) there exists an orientably regular map of type
{
m
,
k
}
with no non-trivial exponents. As an application we show that for every given hyperbolic pair (
k
,
m
) there exist infinitely many orientably regular maps of type
{
m
,
k
}
with no non-trivial exponents, each with the property that the map and its dual have simple underlying graph.
Let
M
be an orientably regular (resp. regular) map with the number
n
vertices. By
G
+
(resp.
G
) we denote the group of all orientation-preserving automorphisms (resp. all automorphisms) of
M
. Let
π
...be the set of prime divisors of
n
. A Hall
π
-subgroup of
G
+
(resp.
G
) is meant a subgroup such that the prime divisors of its order all lie in
π
and the primes of its index all lie outside
π
. It is mainly proved in this paper that (1) suppose that
M
is an orientably regular map where
n
is odd. Then
G
+
is solvable and contains a normal Hall
π
-subgroup; (2) suppose that
M
is a regular map where
n
is odd. Then
G
is solvable if it has no composition factors isomorphic to
PSL
(
2
,
q
)
for any odd prime power
q
≠
3
, and
G
contains a normal Hall
π
-subgroup if and only if it has a normal Hall subgroup of odd order.
Regular and orientably-regular maps are central to the part of topological graph theory concerned with highly symmetric graph embeddings. Classification of such maps often relies on factoring out a ...normal subgroup of automorphisms acting intransitively on the set of the vertices of the map. Maps whose automorphism groups act quasiprimitively on their vertices do not allow for such factorization. Instead, we rely on classification of quasiprimitive group actions which divides such actions into eight types, and we show that four of these types,
HS
,
HC
,
SD
, and
CD
, do not occur as the automorphism groups of regular or orientably-regular maps. We classify regular and orientably-regular maps with automorphism groups of the
HA
type, and construct new families of regular as well as both chiral and reflexible orientably-regular maps with automorphism groups of the
TW
and
PA
types. We provide a brief summary of the known results concerning the
AS
type, which has been extensively studied before.
This is a survey of two related research topics, in each of which surface embeddings of bipartite graphs provide a bridge between two separate areas of mathematics: in the first case these are ...Riemann surface theory and the Galois theory of algebraic number fields; in the second case they are topological graph theory and finite group theory. We give explicit examples to illustrate the connection in each case.
In this paper we classify the reflexible and chiral regular oriented maps with p faces of valency n, and then we compute the asymptotic behaviour of the reflexible to chiral ratio of the regular ...oriented maps with p faces. The limit depends on p and for certain primes p we show that the limit can be 1, greater than 1 and less than 1. In contrast, the reflexible to chiral ratio of regular polyhedra (which are regular maps) with Suzuki automorphism groups, computed by Hubard and Leemans (2014), has produced a nill asymptotic ratio.
We classify those orientably regular maps which are elementary abelian regular branched coverings of Platonic maps
M
, in the case where the covering group and the rotation group
G
of
M
have coprime ...orders. The method involves studying the representations of
G
on certain homology groups of the sphere, punctured at the branch-points. We give a complete classification for branching over faces (or, dually, vertices) of
M
, and we outline how the method extends to other branching patterns.
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