Homework 8
Math 332, Spring 2013
These problems must be written up in LATEX, and are due this Friday, April 5.
0 2
1. (a) Find an eight-element subgroup G of GL(2, Z3 ) containing the matrices
1 0
1 1
and
.
1 2
(b) What is the isomorphism type of G?
2. Let T1 : R3 → R3 be a 180◦ rotation about the line y = −x in the xy-plane, and let
T2 : R3 → R3 be a 120◦ rotation about the line x = y = z (either rotation will do).
(a) Find 3 × 3 matrices A1 and A2 corresponding to the linear transformations T1
and T2 .
(b) Find a six-element subgroup G of GL(3, R) containing A1 and A2 .
(c) What is the isomorphism type of G?
3. Let G be the group of symmetries of the following (flattened) tetrahedron.
H–1, 1, –18L
H1, 1, 18L
H–1, –1, 18L
H1, –1, –18L
(a) List the eight 3 × 3 matrices corresponding to the eight elements of G.
(b) What is the isomorphism type of G?