| This study investigated the set of 3-colorings of the finite cyclic group Z{dollar}sb5{dollar} to define a canonical 3-coloring r for the arbitrary finite larger cyclic groups, generate the 3-colorings for the family of multiples of r, and represent the 3-colorings of the arbitrary finite cyclic groups geometrically through color graphs.; Taking off from Robert Haas' definition, particularly for the 3-coloring r of Z{dollar}sb{lcub}rm m{rcub},{dollar} let n = 3, defined as r:Z{dollar}rmsb{lcub}m{rcub}to Zsb3,{dollar} m {dollar}ge{dollar} 3: Let m = 3k + b, 0 {dollar}le{dollar} b {dollar}<{dollar} 3. Then {dollar}rm r(x)=1-i{dollar} for x {dollar}in{dollar} A{dollar}sb{lcub}rm i{rcub}{dollar}, where{dollar}{dollar}rm Asb{lcub}i{rcub}=cases{lcub}rm{lcub} i(k+1),...,i(k+1)+k{rcub}qquad if 0le i < bcrcrrm{lcub}ik+b,...,ik+b+k-1{rcub}qquad if ble ile 2cr{rcub}{dollar}{dollar}The 3-coloring r of Z{dollar}sb5{dollar} was defined and its multiples were generated through r. The same definition was applied to the larger finite cyclic group Z{dollar}sb{lcub}rm m{rcub}.{dollar} The vertex-coloring procedure was used to represent the 3-colorings of Z{dollar}sb{lcub}rm m{rcub}{dollar} geometrically.; The 3-coloring r of the larger finite cyclic groups followed a simple pattern. Through r, the 3-colorings of the multiples of the cyclic groups were generated. Vertex-coloring can only be applied to some multiples of r, hence not all were represented geometrically.; These 3-colorings of Z{dollar}sb{lcub}rm m{rcub}{dollar} and the topological structure of the graph can be seen in the geometric representation of Z{dollar}sb{lcub}rm m{rcub}.{dollar}; This research may be applied in Abstract Algebra by generating the 3-colorings of finite cyclic groups and of its subgroups. For finite abelian groups, the direct product {dollar}rm Zsb{lcub}m{rcub}times Zsb{lcub}n{rcub},{dollar} 3-colorings could also be studied. |
