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A Book of Abstract Algebra

Charles C. Pinter

Chapter 15

QUOTIENT GROUPS - all with Video Answers

Educators

Section 1

A

01:02

Problem 1

In each of the following, $G$ is a group and $H$ is a normal subgroup of $G$. List the elements of $G / H$ and then write the table of $G / H$.
Example $G=\mathbb{Z}_{6} \quad$ and $\quad H=\{0,3\}$
The elements of $G / H$ are the three cosets $H=H+0=\{0,3\}, H+1=\{1,4\}$, and $H+2=\{2,5\} .$ (Note that $H+3$ is the same as $H+0, H+4$ is the same as $H+1$, and $H+5$ is the same as $H+2$.) The table of $G / H$ is
$$
\begin{array}{c|ccc}
+ & H & H+1 & H+2 \\
\hline H & H & H+1 & H+2 \\
H+1 & H+1 & H+2 & H \\
H+2 & H+2 & H & H+1
\end{array}
$$
$G=\mathbb{Z}_{10}, H=\{0,5\} .\left(\right.$ Explain why $\left.G / H \cong \mathbb{Z}_{5} .\right)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:42

Problem 2

In each of the following, $G$ is a group and $H$ is a normal subgroup of $G$. List the elements of $G / H$ and then write the table of $G / H$.
Example $G=\mathbb{Z}_{6} \quad$ and $\quad H=\{0,3\}$
The elements of $G / H$ are the three cosets $H=H+0=\{0,3\}, H+1=\{1,4\}$, and $H+2=\{2,5\} .$ (Note that $H+3$ is the same as $H+0, H+4$ is the same as $H+1$, and $H+5$ is the same as $H+2$.) The table of $G / H$ is
$$
\begin{array}{c|ccc}
+ & H & H+1 & H+2 \\
\hline H & H & H+1 & H+2 \\
H+1 & H+1 & H+2 & H \\
H+2 & H+2 & H & H+1
\end{array}
$$
$G=S_{3}, H=\{\varepsilon, \beta, \delta\}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:42

Problem 3

In each of the following, $G$ is a group and $H$ is a normal subgroup of $G$. List the elements of $G / H$ and then write the table of $G / H$.
Example $G=\mathbb{Z}_{6} \quad$ and $\quad H=\{0,3\}$
The elements of $G / H$ are the three cosets $H=H+0=\{0,3\}, H+1=\{1,4\}$, and $H+2=\{2,5\} .$ (Note that $H+3$ is the same as $H+0, H+4$ is the same as $H+1$, and $H+5$ is the same as $H+2$.) The table of $G / H$ is
$$
\begin{array}{c|ccc}
+ & H & H+1 & H+2 \\
\hline H & H & H+1 & H+2 \\
H+1 & H+1 & H+2 & H \\
H+2 & H+2 & H & H+1
\end{array}
$$
$G=D_{4}, H=\left\{R_{0}, R_{2}\right\} .$ (See page 68.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:42

Problem 4

In each of the following, $G$ is a group and $H$ is a normal subgroup of $G$. List the elements of $G / H$ and then write the table of $G / H$.
Example $G=\mathbb{Z}_{6} \quad$ and $\quad H=\{0,3\}$
The elements of $G / H$ are the three cosets $H=H+0=\{0,3\}, H+1=\{1,4\}$, and $H+2=\{2,5\} .$ (Note that $H+3$ is the same as $H+0, H+4$ is the same as $H+1$, and $H+5$ is the same as $H+2$.) The table of $G / H$ is
$$
\begin{array}{c|ccc}
+ & H & H+1 & H+2 \\
\hline H & H & H+1 & H+2 \\
H+1 & H+1 & H+2 & H \\
H+2 & H+2 & H & H+1
\end{array}
$$
$G=D_{4}, H=\left\{R_{0}, R_{2}, R_{4}, R_{5}\right\}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:42

Problem 5

In each of the following, $G$ is a group and $H$ is a normal subgroup of $G$. List the elements of $G / H$ and then write the table of $G / H$.
Example $G=\mathbb{Z}_{6} \quad$ and $\quad H=\{0,3\}$
The elements of $G / H$ are the three cosets $H=H+0=\{0,3\}, H+1=\{1,4\}$, and $H+2=\{2,5\} .$ (Note that $H+3$ is the same as $H+0, H+4$ is the same as $H+1$, and $H+5$ is the same as $H+2$.) The table of $G / H$ is
$$
\begin{array}{c|ccc}
+ & H & H+1 & H+2 \\
\hline H & H & H+1 & H+2 \\
H+1 & H+1 & H+2 & H \\
H+2 & H+2 & H & H+1
\end{array}
$$
$G=\mathbb{Z}_{4} \times \mathbb{Z}_{2}, H=\langle(0,1)\rangle=$ the subgroup of $\mathbb{Z}_{4} \times \mathbb{Z}_{2}$ generated by $(0,1)$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:42

Problem 6

In each of the following, $G$ is a group and $H$ is a normal subgroup of $G$. List the elements of $G / H$ and then write the table of $G / H$.
Example $G=\mathbb{Z}_{6} \quad$ and $\quad H=\{0,3\}$
The elements of $G / H$ are the three cosets $H=H+0=\{0,3\}, H+1=\{1,4\}$, and $H+2=\{2,5\} .$ (Note that $H+3$ is the same as $H+0, H+4$ is the same as $H+1$, and $H+5$ is the same as $H+2$.) The table of $G / H$ is
$$
\begin{array}{c|ccc}
+ & H & H+1 & H+2 \\
\hline H & H & H+1 & H+2 \\
H+1 & H+1 & H+2 & H \\
H+2 & H+2 & H & H+1
\end{array}
$$
$G=P_{3}, H=\{\emptyset,\{1\}\} .\left(P_{3}\right.$ is the group of subsets of $\left.\{1,2,3\} .\right)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator