University of Wisconsin-Madison Mathematics Department

Math 541 Modern Algebra A first course in Abstract Algebra Lecturer: Arun Ram

Fall 2007

Homework 10: Due November 15, 2007

To grade: 4, 9, 12, 16.

1. Define $G$-set, stabilizer and orbit.

2. Let $S$ be a $G$-set. Show that the orbits partition $S$.

3. Let $S$ be a $G$-set and let $s\in S$. Show that the stabilizer of $s$ is a subgroup of $G$.

4. Let $S$ be a $G$-set and let $s\in S$. Show that there exists a bijection between $G/G_s$ and $Gs$.

5. Let $S$ be a $G$-set. Let $s\in S$ and $g\in G$. Show that $G_{gs}=g{G}_s{g}^{\mathrm{–1}}$.

6. Let $G$ be a group. The group $G$ acts on itself by left multiplication. Compute the stabilizer and orbit of each element.

7. Define conjugacy class and centralizer and explain the relationship between these and the action of $G$ on itself by conjugation.

8. Let $G$ be a group and let $H$ be a subgroup of $G$. The group $G$ acts on $G/H$ by left multiplication. Compute the stabilizer and orbit of each coset.

9. Define center and conjugacy class and prove the class equation.

10. The symmetric group $S_4$ acts on $S=\{1,2,3,4\}$ by permutations. Compute the stablizer and the orbit of each element.

11. The dihedral group $D_5$ acts on the vertices of a pentagon. Compute the stabilizer and the orbit of each vertex.

12. The dihedral group $D_5$ acts on the edges of a pentagon. Compute the stabilizer and the orbit of each edge.

13. The cyclic group $C_5$ acts on the vertices of a pentagon. Compute the stabilizer and the orbit of each vertex.

14. The cyclic group $C_5$ acts on the edges of a pentagon. Compute the stabilizer and the orbit of each edge.

15. The symmetric group $S_4$ acts on the vertices of a tetrahedron. Compute the stabilizer and the orbit of each vertex.

16. The symmetric group $S_4$ acts on the edges of a tetrahedron. Compute the stabilizer and the orbit of each edge.

17. The symmetric group $S_4$ acts on the faces of a tetrahedron. Compute the stabilizer and the orbit of each face.

18. Describe how the group $(\mathbb{Z}/2\mathbb{Z})\times (\mathbb{Z}/2\mathbb{Z})\times (\mathbb{Z}/2\mathbb{Z})$ acts on the vertices of a cube. Compute the stabilizer and orbit of each vertex.

19. Let $S$ be a $G$-set and let $s\in S$. Show that $\mathrm{Card}\left(G\right)=\mathrm{Card}\left(Gs\right)\mathrm{Card}\left(G_s\right)$.