Question

Do the same for ${\mathrm{\mathbb{Z}}}_{15}$.

Short answer

The ${\mathrm{\mathbb{Z}}}_{15}$can be written as, ${\mathrm{\mathbb{Z}}}_{15}=M\oplus N$.

Step-by-step solution

Subgroups of a group

If A and B are normal subgroups of a group G such that$G=AB$ and , then $G=A\times B$.

Write ℤ15 as a direct sum of two of its subgroup

The${\mathrm{\mathbb{Z}}}_{15}$ is defined as follows:

${\mathrm{\mathbb{Z}}}_{15}=\left\{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14\right\}$

Consider two subgroups $M=\left\{0,3,6,9,12\right\}$and $N=\left\{0,5,10\right\}$.

Then, $M\cap N=\left\{0\right\}$.

Write the direct sum as follows:

$$\begin{gathered} 1=6-5 \\ 2=12-10 \\ 3=3+0 \\ 4=-6+10 \end{gathered}$$

And further find as:

$$\begin{gathered} 5=0-5 \\ 6=6+0 \\ 7=12-5 \\ 8=3+5 \end{gathered}$$

And find calculation further as:

$$\begin{gathered} 9=9+0 \\ 10=0+10 \\ 11=6+5 \\ 12=12+0 \end{gathered}$$

And further simplify as:

$$\begin{gathered} 13=3+10 \\ 14=9+5 \end{gathered}$$

Therefore, ${\mathrm{\mathbb{Z}}}_{15}=M\oplus N$.