Question

Give an example to show that the direct product of cyclic groups need not be cyclic.

Short answer

Answer:

It is proved that the direct product of cyclic groups need not be cyclic.

Step-by-step solution

Cyclic Group

A group $\mathbf{G}$ is called cyclic if there is an element $\mathbf{g}\mathbf{\in }\mathbf{G}$, which can generate the whole group that is,

$$\begin{gathered} \mathbf{G}\mathbf{=}\mathbf{<}\mathbf{g}\mathbf{>} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\{}\mathbf{g}\mathbf{\text{'}\text{'}}\mathbf{/}\mathbf{n}\mathbf{\in }\mathbf{Z}\mathbf{\}} \end{gathered}$$

Show that the direct product of cyclic groups need not be cyclic

Consider two cyclic groups ${\mathrm{\mathbb{Z}}}_2$and ${\mathrm{\mathbb{Z}}}_4$.

The direct product of two cyclic groups ${\mathrm{\mathbb{Z}}}_2$and ${\mathrm{\mathbb{Z}}}_4$is ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4$. The purpose is to show that ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4$is not a cyclic group.

Find the order of ${\mathrm{\mathbb{Z}}}_2$as:

$$\begin{gathered} {\mathrm{\mathbb{Z}}}_2=\left\{0,1\right\} \\ =\left\{1\right\} \end{gathered}$$

The order of 1 in ${\mathrm{\mathbb{Z}}}_2$is 2.

Find the order of ${\mathrm{\mathbb{Z}}}_4$as:

The order of 1 and 3 in ${\mathrm{\mathbb{Z}}}_4$is 4.

Find ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4$as:

${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4=\left\{\left(0,0\right),\left(0,1\right),\left(0,2\right),\left(0,3\right),\left(1,0\right),\left(1,1\right),\left(1,2\right),\left(1,3\right)\right\}$

There does not exist any element in ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4$, which can generate the whole group as there is no element of order 8 in ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4$. The maximum order of element in ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4$is 4.

Therefore, ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4$is the group that is the direct product of two cyclic groups but not cyclic itself.