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- W4386376471 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, <math xmlns=http://www.w3.org/1998/Math/MathML id=M1> <msub> <mrow> <mi>ℤ</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </math> and <math xmlns=http://www.w3.org/1998/Math/MathML id=M2> <msub> <mrow> <mi>ℤ</mi> </mrow> <mrow> <mi>q</mi> </mrow> </msub> </math> where <math xmlns=http://www.w3.org/1998/Math/MathML id=M3> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <msup> <mrow> <mi>ℤ</mi> </mrow> <mrow> <mo>+</mo> </mrow> </msup> </math> and <math xmlns=http://www.w3.org/1998/Math/MathML id=M4> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>≥</mo> <mn>2</mn> </math> . 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 <math xmlns=http://www.w3.org/1998/Math/MathML id=M5> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mi>a</mi> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mi>q</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> </math> and the length of the label is <math xmlns=http://www.w3.org/1998/Math/MathML id=M6> <mi>p</mi> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mi>q</mi> </math> , where <math xmlns=http://www.w3.org/1998/Math/MathML id=M7> <mi>n</mi> </math> is the number of commutator discs." @default.
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- W4386376471 date "2023-08-31" @default.
- W4386376471 modified "2023-09-28" @default.
- W4386376471 title "The Picture on the Presentation of Direct Product Group of Two Cyclic Groups" @default.
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- W4386376471 doi "https://doi.org/10.1155/2023/8018645" @default.
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