Chapter 7, Problem 1E

Let $H=\{0,\pm 3,\pm 6,\pm 9,...\}$ . Find all the left cosets of H in Z.

To determine

To find:

The all left costs of $H$ in $Z$ of the given sets.

Answer to Problem 1E

  $H,1+H,2+H$

Given:

  $H=\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}$

Concept used:

If $G$ be any group and $H$ is nonempty subset of $G$ .

The left-coset of $H$ is $aH=\left\{ah|h\in H\right\}$ For any $a\in G$ .

Calculation:

  $\begin{array}{l}H=\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ H=3\left\{0,\pm 1,\pm 2,\pm 3,\mathrm{.......}\right\}\\ H=3Z\\ H=\left\{3k|k\in Z\right\}\end{array}$

The classes of $nZ$

  $\left\{Z,Z+1,Z+2,\mathrm{......}Z+\left(n-1\right)\right\}$

Therefore, the classes of $3Z$ are $\left\{Z,Z+1,Z+2\right\}$ .

That is

  $\left\{3k|k\in Z\right\},\left\{3k+1|k\in Z\right\},\left\{3k+2|k\in Z\right\}$

Let

  $\begin{array}{l}x=3k,k\in Z\\ x+H=3k+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ 3k\in \left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}=H\\ x+H=3k\end{array}$

Where

  $\left\{aH=Hifa\in H\right\}$

Let

  $\begin{array}{l}x=3k+1,k\in Z\\ x+H=3k+1+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ x+H=1+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ x+H=1+H\end{array}$

Let

  $\begin{array}{l}x=3k+2,k\in Z\\ x+H=3k+2+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ x+H=2+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ x+H=2+H\end{array}$

The left cosets of $H$ in $Z$ are $H,1+H,2+H$ .

Explanation of Solution

Given:

  $H=\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}$

Concept used:

If $G$ be any group and $H$ is nonempty subset of $G$ .

The left-coset of $H$ is $aH=\left\{ah|h\in H\right\}$ For any $a\in G$ .

Calculation:

  $\begin{array}{l}H=\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ H=3\left\{0,\pm 1,\pm 2,\pm 3,\mathrm{.......}\right\}\\ H=3Z\\ H=\left\{3k|k\in Z\right\}\end{array}$

The classes of $nZ$

  $\left\{Z,Z+1,Z+2,\mathrm{......}Z+\left(n-1\right)\right\}$

Therefore, the classes of $3Z$ are $\left\{Z,Z+1,Z+2\right\}$ .

That is

  $\left\{3k|k\in Z\right\},\left\{3k+1|k\in Z\right\},\left\{3k+2|k\in Z\right\}$

Let

  $\begin{array}{l}x=3k,k\in Z\\ x+H=3k+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ 3k\in \left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}=H\\ x+H=3k\end{array}$

Where

  $\left\{aH=Hifa\in H\right\}$

Let

  $\begin{array}{l}x=3k+1,k\in Z\\ x+H=3k+1+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ x+H=1+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ x+H=1+H\end{array}$

Let

  $\begin{array}{l}x=3k+2,k\in Z\\ x+H=3k+2+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ x+H=2+\left\{0,\pm 3,\pm 6,\pm 9,\mathrm{.......}\right\}\\ x+H=2+H\end{array}$

The left cosets of $H$ in $Z$ are $H,1+H,2+H$ .