University of Wisconsin-Madison Mathematics Department

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

Fall 2007

Homework 5: Due October 10, 2007

1. Let $d\in {\mathbb{Z}}_{\ge 0}$. Show that the set $d\mathbb{Z}$ of multiples of $d$ is a subgroup of $\mathbb{Z}$.

2. Show that if $H$ is a subgroup of $\mathbb{Z}$ then there exists a positive integer $d$ such that $H=d\mathbb{Z}$.

3. Let $d_1$ and $d_2$ be positive integers. Show that $d_2\mathbb{Z}\subseteq d_1\mathbb{Z}$ if and only if $d_1$ divides $d_2$.

4. Make a list of all the subgroups of $\mathbb{Z}/110\mathbb{Z}$.

5. Make a list of all the subgroups of the Klein 4 group.

6. Make a list of all the subgroups of the symmetric group $S_3$.