Exercises

1. If $G$ is a group written additively and $S_1$, $S_2$, are complexes we write $S_1 + S_2$ instead of $S_1S_2$. Let $G = \mathbb{Z}$ under $+$. Let $H = \{0, \pm 4, \pm 8, ...\}$ and $K = \{0, \pm 10, \pm 20, ...\}$. Clearly $H \leq G$, $K \leq G$. Determine $H + K$. More generally, if $H = \{na \ \vert\ n \in \mathbb{Z}\}$ and $K = \{nb \ \vert\ n \in \mathbb{Z}\}$. Determine $H + K$. (HINT: Think of the g.c.d.)

2. Complete the proof of Theorem 4.2.1 by showing that if $H_1\leq G$, $H_2 \leq G$ and $H_1H_2 \leq G$, then $H_1H_2 \subset H_2H_1$.

3. Suppose $H$ is a finite nonempty subset of a group $G$ such that $H^2 = HH \subset H$. Prove that $H \leq G$. Is this still true if $H$ is infinite? Why or why not?

4. Consider the following two subgroups of $S_4$: $H = \{1, (1,2)\}$ and $V_4 = \{1, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)\}$. Is $HV_4 \leq S_4$? Why or why not? ($V_4$ is sometimes called the Klein 4-group.)