Chapter 3.1, Problem 51E

Prove that the Cartesian product ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ is an abelian group with respect to the binary operation of addition as defined in Example $11$. (Sec. $3.4,\#27b,$

Sec. $5.1,\#53,$)

Example 11.

Consider the additive groups ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$. To avoid any unnecessary confusion we write ${\left[a\right]}_2$ and ${\left[a\right]}_4$ to designate elements in ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$, respectively. The Cartesian product of ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$ can be expressed as

${\mathbb{Z}}_2\times {\mathbb{Z}}_4=\left\{\left({\left[a\right]}_2,{\left[b\right]}_4\right)|{\left[a\right]}_2\in {\mathbb{Z}}_2,{\left[b\right]}_4\in {\mathbb{Z}}_4\right\}$

Sec. $3.4,\#27b\ll$

27. Prove or disprove that each of the following groups with addition as defined in Exercises $52$ of section $3.1$ is cyclic.

a. ${\mathbb{Z}}_2\times {\mathbb{Z}}_3$

b. ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$

Sec. $5.1,\#53\ll$

53. Rework Exercise $52$ with the direct sum ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_4$.