1. Recall that the group of rigid motions of the tetrahedron and the alternating group A₄ are isomorphic.

1. What is the product of two rotations of order 3 about different axes? (Try rotations by the same angle and rotations by different angles).
2. Verify your results algebraically.
3. By finding a proper normal subgroup, show that A₄ is not simple.
4. In proving that A₅ is simple, we used products of rotations of the dodecahedron. Why does a similar proof not work here?

2. Let R[x,y,z] be the set of real polynomials in three variables with the usual addition and multiplication. Then S₃ acts on R[x,y,z] by permuting variables, for example if a = (132) then p(x,y,z)^a = p(z,x,y) .

1. Show that this is a group action.
2. Show that for all p, q in R[x,y,z], and all a in S₃, (p+q)^a = p^a + q^a and (pq)^a = p^a q^a.
3. Find three elements of the kernel of the action, and describe this kernel.

3. Both GL(3,R) and O(3,R) act on R³ by right multiplication of row vectors. Let 0 not = v in R³ .

1. Determine the fixing groups O(3,R)_v and $GL(3,R)_v .
2. Determine the orbits of v under O(3,R) and GL(3,R).
3. How many orbits do O(3,R) and GL(3,R) have?




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 Last update May 7, 2000

 Author: Phill Schultz, schultz@maths.uwa.edu.au