| Summary: | This article aims to create commutative hyperstructures, starting with a non-commutative group. Therefore, we consider the starting group to be the dihedral group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>n</mi></msub></semantics></math></inline-formula>, where <i>n</i> is a natural number, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, and we determine the HX groups associated with the dihedral group. For a fixed number <i>n</i>, we note <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">G</mi><mi>n</mi></msub><mo>=</mo><mfenced separators="" open="{" close="}"><msubsup><mi mathvariant="script">G</mi><mrow><msub><mi>p</mi><mn>2</mn></msub></mrow><msub><mi>p</mi><mn>1</mn></msub></msubsup><mspace width="4.pt"></mspace><mi>H</mi><mi>X</mi><mo>−</mo><mrow><mi>groups</mi><mo>,</mo></mrow><mspace width="4.pt"></mspace><mi>for</mi><mspace width="4.pt"></mspace><mi>any</mi><mspace width="4.pt"></mspace><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><mspace width="4.pt"></mspace><msub><mi>p</mi><mn>2</mn></msub><mo>∈</mo><msup><mi mathvariant="double-struck">N</mi><mo>*</mo></msup><mspace width="4.pt"></mspace><mi>such</mi><mspace width="4.pt"></mspace><mi>that</mi><mspace width="4.pt"></mspace><mi>n</mi><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>2</mn></msub></mfenced></mrow></semantics></math></inline-formula> as the set of all HX groups. This paper analyses this new structure’s properties for particular cases when the dihedral group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mn>4</mn></msub></semantics></math></inline-formula> is the support group.
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