5 research outputs found
Regular maps with nilpotent automorphism groups
AbstractWe study regular maps with nilpotent automorphism groups in detail. We prove that every nilpotent regular map decomposes into a direct product of maps H×K, where Aut(H) is a 2-group and K is a map with a single vertex and an odd number of semiedges. Many important properties of nilpotent maps follow from this canonical decomposition, including restrictions on the valency, covalency, and the number of edges. We also show that, apart from two well-defined classes of maps on at most two vertices and their duals, every nilpotent regular map has both its valency and covalency divisible by 4. Finally, we give a complete classification of nilpotent regular maps of nilpotency class 2
Orientably-Regular -Maps and Regular -Maps
Given a map with underlying graph , if the set of prime divisors
of is denoted by , then we call the map a {\it
-map}.
An orientably-regular (resp. A regular ) -map is called {\it solvable}
if the group of all orientation-preserving automorphisms (resp. the group
of automorphisms) is solvable; and called {\it normal} if (resp. )
contains a normal -Hall subgroup.
In this paper, it will be proved that orientably-regular -maps are
solvable and normal if and regular -maps are solvable if
and has no sections isomorphic to for some
prime power . In particular, it's shown that a regular -map with
is normal if and only if is isomorphic to a
Sylow -group of .
Moreover, nonnormal -maps will be characterized and some properties and
constructions of normal -maps will be given in respective sections.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:2201.0430
Regular embeddings of cycles with multiple edges revisited
Regularne vložitve ciklov z večkratnimi povezavami se pojavljajo v literaturi že kar nekaj časa, tako v topološki teoriji grafov kot tudi izven nje. Ta članek izriše kompletno podobo teh zemljevidov na ta način, da povsem opiše, klasificira in enumerira regularne vložitve ciklov z večkratnimi povezavami tako na orientabilnih kot tudi na neorientabilnih ploskvah. Večina rezultatov je sicer znana v tej ali oni obliki, toda tu so predstavljeni iz poenotenega zornega kota, osnovanega na teoriji končnih grup. Naš pristop daje dodatno informacijo tako o zemljevidih kot o njihovih grupah avtomorfizmov, priskrbi pa tudi dodaten vpogled v njihove odnose.Regular embeddings of cycles with multiple edges have been reappearing in the literature for quite some time, both in and outside topological graph theory. The present paper aims to draw a complete picture of these maps by providing a detailed description, classification, and enumeration of regular embeddings of cycles with multiple edges on both orientable and non-orientable surfaces. Most of the results have been known in one form or another, but here they are presented from a unique viewpoint based on finite group theory. Our approach brings additional information about both the maps and their automorphism groups, and also gives extra insight into their relationships