C Roettger Spring 2013 Math 301 – Homework 6 Problem 7.46 The group D4 acts as a group of permutations of the square with vertices (0, 0), (2, 0), (0, 2), (2, 2) (all symmetries are about the center (1, 1)). Sketch this region five times and indicate in each sketch the orbit of one of the points A = (2, 1), B = (2, 2), C = (1, 0), D = (2, 1.8), E = (1, 0.3). Also indicate the size of the stabilizer in D4 for each point. Problem 7.48b Find the order of the group of rotations of a regular octahedron (a solid with eight congruent equilateral triangles as faces). By the way, it’s great fun to build an octahedron by drawing eight connected equilateral triangles and cutting them out/folding and gluing them together! Problem 8.6 Prove, by comparing orders of elements, that Z8 ⊕ Z2 is not isomorphic to Z4 ⊕ Z4 . Problem 8.20 Determine the number of elements of order 15 and the number of cyclic subgroups of order 15 in G = Z30 ⊕ Z20 .