<![CDATA[The set of all plane geometric transformations forms a group under the binary operation of function composition. One of its subgroups consists of all transformations that can be expressed in the form T(x) = Ax + v, where A is an invertible (2x2) and v is a fixed vector . This study aims to identify the existence and structure of certain subgroups within through a linear algebra approach. The research methods include a literature review, simulations on specific cases to obtain a more concrete understanding of the problem, and deductive reasoning based on mathematical syllogisms to derive properties and theorems that can be algebraically verified. Consistent with the research objectives, the concepts and theoretical foundations employed are drawn from the analytical properties of plane geometry and linear algebra. These concepts and theorems are revisited to ensure their relevance to the research problem and applicability in its resolution. By applying these theoretical constructs to the problem, several subgroups whose existence can be proven algebraically are identified. These subgroups include the translation subgroup, the subgroup containing rotation transformations, the group of isometries, and the group of similarities. ]]>
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