Chapter 14, HOMOMORPHISMS Video Solutions, A Book of Abstract Algebra

Problem 1

Consider the function $f: \mathbb{Z}_{8} \rightarrow \mathbb{Z}_{4}$ given by
$$
f=\left(\begin{array}{llllllll}
0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
0 & 1 & 2 & 3 & 0 & 1 & 2 & 3
\end{array}\right)
$$
Verify that $f$ is a homomorphism, find its kernel $K$, and list the cosets of $K$. [REMARK: To verify that $f$ is a homomorphism, you must show that $f(a+b)=$ $f(a)+f(b)$ for all choices of $a$ and $b$ in $\mathbb{Z}_{8}$; there are 64 choices. This may be accomplished by checking that $f$ transforms the table of $\mathbb{Z}_{g}$ to the table of $\mathbb{Z}_{4}$, as on page $132 .$ ]

James Kiss

Numerade Educator

08:50

Problem 2

Consider the function $f: S_{3} \rightarrow \mathbb{Z}_{2}$ given by
$$
f=\left(\begin{array}{cccccc}
\varepsilon & \alpha & \beta & \gamma & \delta & \kappa \\
0 & 1 & 0 & 1 & 0 & 1
\end{array}\right)
$$
Verify that $f$ is a homomorphism, find its kernel $K$, and list the cosets of $K$.

Ely Crowder

Numerade Educator

08:50

Problem 3

Find a homomorphism $f: \mathbb{Z}_{15} \rightarrow \mathbb{Z}_{5}$, and indicate its kernel. (Do not actually verify that $f$ is a homomorphism.)

Ely Crowder

Numerade Educator

08:50

Problem 4

Imagine a square as a piece of paper lying on a table. The side facing you is side
A. The side hidden from view is side $B$. Every motion of the square either inter-
changes the two sides (that is, side $B$ becomes visible and side $A$ hidden) or leaves the sides as they were. In other words, every motion $R_{i}$ of the square brings about one of the permutations
$$
\left(\begin{array}{ll}
A & B \\
A & B
\end{array}\right) \quad \text { or } \quad\left(\begin{array}{ll}
A & B \\
B & A
\end{array}\right)
$$
of the sides; call it $g\left(R_{i}\right)$. Verify that $g: D_{4} \rightarrow S_{2}$ is a homomorphism, and give its kernel.

Ely Crowder

Numerade Educator

00:17

Problem 5

Every motion of the regular hexagon brings about a permutation of its diagonals, labeled 1,2, and $3 .$ For each $R_{i} \in D_{6}$, let $f\left(R_{i}\right)$ be the permutation of the diagonals
produced by $R_{i} .$ Argue informally (appealing to geometric intuition) to explain why $f: D_{6} \rightarrow S_{3}$ is a homomorphism. Then complete the following:
$$
f\left(\begin{array}{llllll}
1 & 2 & 3 & 4 & 5 & 6 \\
1 & 2 & 3 & 4 & 5 & 6
\end{array}\right)=\varepsilon \quad f\left(\begin{array}{cccccc}
1 & 2 & 3 & 4 & 5 & 6 \\
2 & 3 & 4 & 5 & 6 & 1
\end{array}\right)=\delta \quad \ldots
$$
(That is, find the value of $f$ on all 12 elements of $D_{6} .$ )

Ashley High

Numerade Educator

02:05

Problem 6

Let $B \subset A .$ Let $h: P_{A} \rightarrow P_{B}$ be defined by $h(C)=C \cap B .$ For $A=\{1,2,3\}$ and $B=\{1,2\}$, complete the following:
$$
h=\left(\begin{array}{llllll}
\emptyset & \{1\} \quad\{2\} \quad\{3\} & \{1,2\} & \{1,3\} & \{2,3\} & A
\end{array}\right)
$$
For any $A$ and $B \subset A$, argue as in Chapter 3, Exercise $\mathrm{C}$ to show that $h$ is a homomorphism.

Varsha Aggarwal

Numerade Educator