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

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  • Yanita Yanita
  • Budi Rudianto
  • Kannan Krithivasan

Abstract

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, ℤp and ℤq where p,q∈ℤ+ and p,q≥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 aq−1bnabq−n and the length of the label is p+2n−q, where n is the number of commutator discs.

Suggested Citation

  • Yanita Yanita & Budi Rudianto & Kannan Krithivasan, 2023. "The Picture on the Presentation of Direct Product Group of Two Cyclic Groups," Journal of Applied Mathematics, Hindawi, vol. 2023, pages 1-6, August.
    • Handle: RePEc:hin:jnljam:8018645
    DOI: 10.1155/2023/8018645
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