University of Wisconsin-Madison Mathematics Department

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

Fall 2007

Homework 11: Due November 21, 2007

To grade: 5, 10, 11, 13, 14, 17.

1. Let $S$ be a subset of a group $G$. Define the subgroup generated by $S$.

2. Let $S$ be a subset of a ring $R$. Define the ideal generated by $S$.

3. Let $S$ be a subset of a module $M$. Define the submodule generated by $S$.

4. Let $S$ be a subset of a vector space $V$. Define the subspace generated by $S$.

5. Let $S$ be a subset of a vector space $V$. Show that $\mathrm{span}\left(S\right)$ is equal to the set of linear combinations of elements of $S$.

6. Show that the intersection of two subgroups of a group $G$ is a subgroup of $G$.

7. Give an example to show that the union of two subgroups of $G$ is not necessarily a subgroup of $G$.

8. Let $G$ be a group and let $S$ be a subset of $G$. Let $\mathscr{H}$ be the set of subgroups $H$ of $G$ such that $S\subseteq H$. Define

   $H_S={\bigcap }_{H\in \mathscr{H}}H$

   1. Show that $H_S$ is a subgroup of $G$.
   2. Show that $S\subseteq H_S$.
   3. Show that if $H$ is a subgroup of $G$ and $S\subseteq H$ then $H_S\subseteq H$.
   Conclude that $H_S=⟨S⟩$.

9. Determine the subgroup lattice of the dihedral group $D_4$. The group $D_4$ acts on its subgroups by conjugation. Determine the stabilizer and the orbit of each subgroup.

10. Determine the subgroup lattice of the quaternion group $Q$. The group $Q$ acts on its subgroups by conjugation. Determine the stabilizer and the orbit of each subgroup.

11. Determine the subgroup lattice of the dihedral group $D_5$. The group $D_5$ acts on its subgroups by conjugation. Determine the stabilizer and the orbit of each subgroup.

12. The dihedral group $D_4$ acts on its elements by conjugation. Determine the stabilizer and the orbit of each element. Determine the conjugacy classes of $D_4$, the centralizer of each element, and determine the center of $D_4$.

13. The quaternion group $Q$ acts on its elements by conjugation. Determine the stabilizer and the orbit of each element. Determine the conjugacy classes of $Q$, the centralizer of each element, and determine the center of $Q$.

14. The dihedral group $D_5$ acts on its elements by conjugation. Determine the stabilizer and the orbit of each element. Determine the conjugacy classes of $D_5$, the centralizer of each element, and determine the center of $D_5$.

15. Determine the subgroup lattice of the quaternion group $Q$. The group $Q$ acts on its subgroups by conjugation. Determine the stabilizer and the orbit of each subgroup.

16. Let $A$ be a matrix. Explain how to use $A$ to produce a module for the ring $ℂ\left[x\right]$.
17. Use the matrix

    $A=\left(\begin{array}{ccc}1& -2& -1\\ -3& 6& 3\\ 6& -12& -6\end{array}\right)$

    to define a $ℂ\left[x\right]$-module on the vector space $V$ with basis $b_1$, $b_2$, $b_3$. Compute $(x^2+2x+1)b_2$.
18. Find all submodules of the module in Problem 17.
19. Let $V$ be the vector space with basis $b_1$, $b_2$, $b_3$. The matrix

    $A=\left(\begin{array}{ccc}1& -2& -1\\ -3& 6& 3\\ 6& -12& -6\end{array}\right)$

    defines a linear transformation $f:V\to V$. Find $\ker \hspace{0.17em}f$.