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 .