Pries: M467 - Abstract Algebra I, Spring 2013
Homework 1: Group actions and symmetry
Read: Judson pages 40-42, 76-89, 213-216
Problems:
1. Consider the action of the dihedral group D6 on the vertices of a regular hexagon.
(a) Use this to find an injective homomorphism D6 ,→ S6 .
(b) Use this to find an injective homomorphism D6 ,→ GL2 (R).
(c) Find the order of each symmetry (the smallest positive integer e such that g e = id).
2. Let G be the group of rotational symmetries of a tetrahedron.
(a) What is the order of G? Explain your answer.
(b) Prove that there is injective homomorphism φ : G ,→ S4 .
(c) Prove that G ' A4 .
3. Judson pg 91-92 #11, 27, 29.
4. Judson pg 227-229 #4, 5, 22.