The Picture on the Presentation of Direct Product Group of Two Cyclic Groups

A picture in a group presentation is a geometric configuration with an arrangement of discs and arcs within a boundary disc. The drawing of this picture does not have to follow a particular rule, only using the generator as discs and the relation as arcs. It will form a picture label pattern if drawn with a particular rule. This paper discusses the label pattern of a picture in the presentation of direct product groups. Direct product presentation is used with two cyclic groups, ${\mathbb{Z}}_p$ and ${\mathbb{Z}}_q$ where $p,q\in {\mathbb{Z}}^{+}$ and $p,q\ge 2$ . The method for forming a picture label pattern is to arrange the first generator in the initial arrangement, compile a second generator, and add a number of commutators. Furthermore, the pattern is used to calculate the length of the label on the picture. It is obtained that the picture’s label is $a^{q-1}{b}^{n}a{b}^{q-n}$ and the length of the label is $p+2n-q$ , where $n$ is the number of commutator discs.