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

Yanita Yanita and Budi Rudianto

Journal of Applied Mathematics, 2023, vol. 2023, issue 1

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.

Date: 2023
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https://doi.org/10.1155/2023/8018645

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2023:y:2023:i:1:n:8018645

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