Pries: M466 - Abstract Algebra I, Fall 2014
Week 1: Symmetries and permutations
Due Friday August 29
Read: Judson pages 40-42, Chapter 5 (76-89)
Discussion Problems:
1. Explain why the symmetry group of a water molecule H2 O is Z/2 × Z/2
(hint - include reflections).
2. Analyze D6 , the symmetry group of a regular hexagon.
(a) Find the order of each symmetry (the smallest positive integer e such that g e = id).
(b) Find the center of D6 (a symmetry g such that gh = hg for all symmetries h).
3. Describe the symmetries r and s in Dn using (a) permutations and (b) matrices.
4. Judson pg 93, 36a: prove that srs−1 = r−1 in Dn .
5. Analyze the symmetry group of a tetrahedron.
(a) How many rotational symmetries are there?
(b) Label the 4 vertices and write down the permutation for each symmetry.
(c) Describe the symmetry group as a subgroup of the symmetric group S4 .
(d) What changes if you include reflection symmetries?