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Abstract

One of the simplest classes of finite groups used as a source of counterexamples in a first course of modern algebra is the class of finite dihedral groups. Among the subgroups of dihedral group, finding subgroups of index 2 is of interest in part because these subgroups are normal subgroups. In this article, we use the representations of the symmetries of the dihedral groups as permutations of the vertices and determine concretely all its subgroups of index 2. Under this representation or embedding, the article determines the intersection of the dihedral group with the corresponding alternating groups when they are naturally viewed as subgroup of the symmetry group on the set of vertices of the regular n-gon. In its second part, the article considers the question of embedding an arbitrary finite group into a symmetry group using the well-known Cayley embedding. In this more general context where every element of the group is viewed as a permutation, one counts the even elements of the group.

Author Bio

The author is an undergraduate student majoring in Mathematics at University of Houston Downtown. She is driven by an insatiable curiosity to explore the beauty of mathematical theory and its practical applications. In addition to her academic coursework, she has actively participated in research endeavors. During the rising year from junior to senior, after the first course of abstract algebra, the professor offered her an opportunity to work with him on a project, in which they would like to count the number of even symmetries of the regular polygon and to specify how these groups are expressed in terms of permutation of vertices. For her, it opened the door into the world of abstract algebra and conducting the research in mathematics. Besides abstract algebra, she is also interested in machine learning, biological mathematics, math modeling and partial differential equations. Looking ahead, she is eager to continue my academic journey and pursue opportunities that allow her to contribute meaningfully to the field of mathematics. Specifically, she is going to apply for a postgraduate program in Mathematics and Statistics after graduation from University of Houston Downtown.

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