Semiregular Gloria Map

Even though regular maps are historically at the center of study of highly symmet-ric maps, there are also many reasons for studying maps that do not possess highestlevels of symmetry. While the main reason for relaxing the symmetry conditions putupon the regular maps may be seen in providing further insights into the requirementsand properties leading to regularity, non-regular maps are arguably very interesting ontheir own. Due to the action of the automorphism groups of maps being necessar-ilysemi-regular on darts, maps whose (orientation preserving) automorphism groups actvia two independent orbits of darts are the closest to being regular, and as such serveas a natural gateway to the study of both regular and non-regular maps.A full classification of half-regular Cayley maps using the concept of kew-morphismshas been recently obtained by Jajcay and Nedela. They have shown that half-regularCayley maps come in two types: those that arise from two skew-morphism orbits ofequal size that are both closed under inverses and those that arise from two equal-sizedorbits that do not contain involutions or inverses but one contains the inverses of theother. In addition, half-regular Cayley maps of the first type were shown to be half-edge-transitive and half-regular Cayley maps of the second type were shown to be necessarilyedge-transitive.Based on the successful study of half-regularity, it is interesting to observe that thisconcept can be further extended and generalized to include the study of Cayley mapswhose dart sets partition into even more subsets under the action of their automorphismgroup. It appears quite feasible that the existence of such Cayley maps is related toskew-morphisms again (skew-morphisms with more than two orbits that combine intoa generating set for the respective Cayley group).The second section of this thesis mainly use MAGMA to calculate half-regularCayley maps of cyclic groups. We use cyclic group of order15as example to illustratethe process on calculating half-regular Cayley maps. we make details on the orbits of itsskew-morphisms and the corresponding half-regular Cayley maps. In the third sectionof this article, we also research the non-formal and the formal nature of the cyclic group as a preparation for the further study on half-regular Cayley maps of non-cyclic groups.