A Number-Theoretic Approach to Subgroups of Dihedral Groups
AIR FORCE ACADEMY COLORADO SPRINGS CO
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Dihedral groups describe the symmetry of bounded figures in the plane. They are also the fundamental structures used to analyze the symmetry of more complicated objects like frieze patterns, wallpaper patterns, and three- dimensional crystals. This report investigates the subgroup structure of dihedral groups from a number theory point of view. The prerequisites needed to understand the material are modest. It is assumed that the reader is familiar with the group theory and number theory at the undergraduate level. The report presents the authors research findings from August 1989 to March 1990. During that period, two papers were submitted for publication they are reproduced here in chapters one and two. Footnotes have been added to clarify and expand the topics covered. Also, for those wishing to pursue the subject further, a section on open questions has been included, as well as an extended bibliography. The footnotes, open questions, and extended bibliography unify what otherwise might appear to be two disjoint papers.
- Theoretical Mathematics