81 5 121 126 n Sn 58 4 90 94 τ(n) σ(n) 82 4 126 130 59 2 60 62 35 4 48 52 83 2 84 86 36 9 91 100 60 12 168 180 37 2 38 40 61 2 62 64 84 12 224 236 38 4 60 64 62 4 96 100 85 4 108 112 63 6 104 110 86 4 132 136 39 4 56 60 64 7 127 134 87 4 120 124 40 8 90 98 88 8 180 188 n Sn 41 2 42 44 τ(n) σ(n) 42 8 96 104 65 4 84 88 89 2 90 92 43 2 44 46 66 8 144 152 90 12 234 246 44 6 84 90 67 2 68 70 91 4 112 116 92 6 168 174 45 6 78 84 68 6 126 132 93 4 128 132 69 4 96 100 46 4 72 76 94 4 144 148 47 2 48 50 70 8 144 152 48 10 124 134 71 2 72 74 n Sn τ(n) σ(n) 49 3 57 60 72 12 195 207 95 4 120 124 50 6 93 99 73 2 74 76 96 12 252 264 74 4 114 118 51 4 72 76 n Sn τ(n) σ(n) 75 6 124 130 52 6 98 104 97 2 98 100 53 2 54 56 76 6 140 146 98 6 171 177 54 8 120 128 77 4 96 100 n Sn τ(n) σ(n) 55 4 72 76 78 8 168 176 99 6 156 162 56 8 120 128 79 2 80 82 100 9 217 226 57 4 80 84 80 10 186 196 Bringing the Dihedral Groups D(n) into the Classroom An interesting aspect of this project is that I take the geometric concepts of the symmetries of regular polygons (all of which are the elements of D(n)) and utilize abstract algebra by investigating the number of subgroups of the dihedral group D(n). The research closes nicely with a formula that utilizes basic number theory properties. Rather than applying abstract algebra and number theory concepts, geometric principles will be applied throughout the innovation of this project. The overarching goal of this project is to allow high school students the opportunity to work with complex mathematical topics that are more commonly found in undergraduate courses.
n τ(n) σ(n) Sn 35 4 48 52 36 9 91 100 37 2 38 40 38 4 60 64 39 4 56 60 40 8 90 98 41 2 42 44 42 8 96 104 43 2 44 46 44 6 84 90 45 6 78 84 46 4 72 76 47 2 48 50 48 10 124 134 49 3 57 60 50 6 93 99 51 4 72 76 52 6 98 104 53 2 54 56 54 8 120 128 55 4 72 76 56 8 120 128 57 4 80 84 58 4 90 94 59 2 60 62 60 12 168 180 61 2 62 64 62 4 96 100 63 6 104 110 64 7 127 134 n τ(n) σ(n) Sn 65 4 84 88 66 8 144 152 67 2 68 70 68 6 126 132 69 4 96 100 70 8 144 152 71 2 72 74 72 12 195 207 73 2 74 76 74 4 114 118 75 6 124 130 76 6 140 146 77 4 96 100 78 8 168 176 79 2 80 82 80 10 186 196 81 5 121 126 82 4 126 130 83 2 84 86 84 12 224 236 85 4 108 112 86 4 132 136 87 4 120 124 88 8 180 188 89 2 90 92 90 12 234 246 91 4 112 116 92 6 168 174 93 4 128 132 94 4 144 148 n τ(n) σ(n) Sn 95 4 120 124 96 12 252 264 n τ(n) σ(n) Sn 97 2 98 100 98 6 171 177 n τ(n) σ(n) Sn 99 6 156 162 100 9 217 226 Bringing the Dihedral Groups D(n) into the Classroom An interesting aspect of this project is that I take the geometric concepts of the symmetries of regular polygons (all of which are the elements of D(n)) and utilize abstract algebra by investigating the number of subgroups of the dihedral group D(n). The research closes nicely with a formula that utilizes basic number theory properties. Rather than applying abstract algebra and number theory concepts, geometric principles will be applied throughout the innovation of this project. The overarching goal of this project is to allow high school students the opportunity to work with complex mathematical topics that are more commonly found in undergraduate courses.