Contemporary Abstract Algebra

Joseph Gallian

Chapter 26 Generators and Relations - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $S$ be a set of distinct symbols. Show that the relation defined on $W(S)$ in this chapter is an equivalence relation.

Anurag Kumar

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01:40

Problem 2

Let $n$ be an even integer. Prove that $D_{n} / Z\left(D_{n}\right)$ is isomorphic to $D_{n / 2}$.

Adriano Chikande

Numerade Educator

00:31

Problem 3

Verify that the set $K$ in Example 2 is closed under multiplication on the left by $b$.

Amrita Bhasin

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03:58

Problem 4

Show that $\left\langle a, b \mid a^{5}=b^{2}=e, b a=a^{2} b\right\rangle$ is isomorphic to $Z_{2}$.

Anthony Ramos

Numerade Educator

01:07

Problem 5

Prove Theorem $26.3$ and its corollary.

Manik Pulyani

Numerade Educator

05:10

Let $G$ be the group $\{\pm 1, \pm i, \pm j, \pm k\}$ with multiplication defined as in Exercise 54 in Chapter 9 . Show that $G$ is isomorphic to $\langle a, b|$ $\left.a^{2}=b^{2}=(a b)^{2}\right\rangle .$ (Hence, the name "quaternions.")

Ely Crowder

Numerade Educator

05:16

Problem 7

In any group, show that $\langle a, b\rangle=\langle a, a b\rangle .$ (This exercise is referred to in the proof of Theorem $26.5$.)

Amit Srivastava

Numerade Educator

08:50

Problem 8

Let $\alpha=(12)(34)$ and $\beta=(24)$. Show that the group generated by $\alpha$ and $\beta$ is isomorphic to $D_{4}$.

Ely Crowder

Numerade Educator

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Problem 9

Prove that $G=\left\langle x, y \mid x^{2}=y^{n}=e, x y x=y^{-1}\right\rangle$ is isomorphic to $D_{n^{*}}$ (This exercise is referred to in the proof of Theorem $26.5 .)$

Nick Johnson

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01:50

Problem 10

What is the minimum number of generators needed for $Z_{2} \oplus Z_{2} \oplus$ $Z_{2}$ ? Find a set of generators and relations for this group.

Varsha Aggarwal

Numerade Educator

06:11

Problem 11

Suppose that $x^{2}=y^{2}=e$ and $y z=z x y .$ Show that $x y=y x$.

Stephanie Carter

Numerade Educator

01:39

Problem 12

Let $G=\left\langle a, b \mid a^{2}=b^{4}=e, a b=b^{3} a\right\rangle .$
a. Express $a^{3} b^{2} a b a b^{3}$ in the form $b^{i} a^{j}$, where $0 \leq i \leq 1$ and $0 \leq j \leq 3$
b. Express $b^{3} a b a b^{3} a$ in the form $b^{\prime} a^{\prime}$, where $0 \leq i \leq 1$ and $0 \leq j \leq 3$.

Gaurav Kalra

Numerade Educator

02:59

Problem 13

Let $G=\left\langle a, b \mid a^{2}=b^{2}=(a b)^{2}\right\rangle$.
a. Express $b^{2} a b a b^{3}$ in the form $b^{i} a^{j}$.
b. Express $b^{3} a b a b^{3} a$ in the form $b^{l} a^{\prime}$.

Alekhya Bhupalam

Numerade Educator

01:07

Problem 14

Let $G$ be the group defined by the following table. Show that $G$ is isomorphic to $\bar{D}_{n^{-}}$

Varsha Aggarwal

Numerade Educator

02:07

Problem 15

Let $G=\left\langle x, y \mid x^{8}=y^{2}=e, y x y x^{3}=e\right\rangle .$ Show that $|G| \leq 16$. Assuming that $|G|=16$, find the center of $G$ and the order of $x y$.

James Kiss

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00:57

Problem 16

Confirm the classification given in Table $26.1$ of all groups of orders 1 to 11 .

Rithvik Manne

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01:51

Problem 17

Let $G$ be defined by some set of generators and relations. Show that every factor group of $G$ satisfies the relations defining $G$.

Varsha Aggarwal

Numerade Educator

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Problem 18

Let $G=\langle s, t|$ sts $=t s t\rangle .$ Show that the permutations $(23)$ and (13) satisfy the defining relations of $G$. Explain why this proves that $G$ is non-Abelian.

Nick Johnson

Numerade Educator

01:02

Problem 19

In $D_{12}=\left\langle x, y \mid x^{2}=y^{12}=e, x y x=y^{-1}\right\rangle$, prove that the subgroup $H=\left\langle x, y^{3}\right\rangle$ (which is isomorphic to $D_{4}$ ) is not a normal subgroup.

Varsha Aggarwal

Numerade Educator

01:40

Problem 20

Let $G=\left\langle x, y \mid x^{2 n}=e, x^{n}=y^{2}, y^{-1} x y=x^{-1}\right\rangle .$ Show that $Z(G)=$
$\left\{e, x^{n}\right\} .$ Assuming that $|G|=4 n$, show that $G / Z(G)$ is isomorphic to $D_{n} .$ (The group $G$ is called the dicyclic group of order $4 n$.)

Varsha Aggarwal

Numerade Educator

01:57

Problem 21

Let $G=\left\langle a, b \mid a^{6}=b^{3}=e, b^{-1} a b=a^{3}\right\rangle .$ How many elements
does $G$ have? To what familiar group is $G$ isomorphic?

Wendi Zhao

Numerade Educator

01:36

Problem 22

Let $G=\left\langle x, y \mid x^{4}=y^{4}=e, x y x y^{-1}=e\right\rangle .$ Show that $|G| \leq 16 .$ As-
suming that $I G \mid=16$, find the center of $G$ and show that $G /\left\langle y^{2}\right\rangle$ is isomorphic to $D_{4}$.

Varsha Aggarwal

Numerade Educator

01:42

Problem 23

Determine the orders of the elements of $D_{\infty}$

Nick Johnson

Numerade Educator

03:58

Problem 24

$$
\begin{aligned}
&\text { Let } G=\left\{\left[\begin{array}{lll}
1 & a & b \\
0 & 1 & c \\
0 & 0 & 1
\end{array}\right] \mid a, b, c \in Z_{2}\right\} \text { . Prove that } G \text { is isomorphic }\\
&\text { to } D_{4} \text { . }
\end{aligned}
$$

Anthony Ramos

Numerade Educator

02:04