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A First Course in Abstract Algebra

John B. Fraleigh; Victor J. Katz

Chapter 7

Generating Sets and Cayley Digraphs - all with Video Answers

Educators

Chapter Questions

01:32

Problem 1

List the elements of the subgroup generated by the given subset.The subset $\{2,3\}$ of $\mathbb{Z}_{12}$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:32

Problem 2

List the elements of the subgroup generated by the given subset.The subset $\{4,6\}$ of $\mathbb{Z}_{12}$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
00:33

Problem 3

List the elements of the subgroup generated by the given subset.The subset $\{4,6\}$ in $\mathbb{Z}_{25}$

Jeyasree R T
Jeyasree R T
Numerade Educator
02:04

Problem 4

List the elements of the subgroup generated by the given subset.The subset $\{12,30\}$ of $\mathbb{Z}_{36}$

David Collins
David Collins
Numerade Educator
01:40

Problem 5

List the elements of the subgroup generated by the given subset.The subset $\{12,42\}$ of $\mathbb{Z}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:29

Problem 6

List the elements of the subgroup generated by the given subset.The subset $\{18,24,39\}$ of $\mathbb{Z}$

Carlene Jimenez
Carlene Jimenez
Numerade Educator
01:40

Problem 7

List the elements of the subgroup generated by the given subset.The subset $\left\{\mu, \mu \rho^{2}\right\}$ in $D_{8}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:04

Problem 8

List the elements of the subgroup generated by the given subset.The subset $\left\{\rho^{8}, \rho^{10}\right\}$ in $D_{18}$

David Collins
David Collins
Numerade Educator
04:00

Problem 9

Use the Cayley digraph in Figure $7.15$ to compute these products. Note that the solid edges represent the generator $a$ and the dashed lines represent $b$.
a. $\left(b a^{2}\right) a^{3}$
b. $(b a)\left(b a^{3}\right)$
c. $b\left(a^{2} b\right)$

Jose Hannan
Jose Hannan
Numerade Educator
02:42

Problem 10

Give the table for the group having the indicated digraph. In each digraph, take $e$ as identity element. List the identity $e$ first in your table, and list the remaining elements alphabetically, so that your answers will be easy to check.The digraph in Fig. $7.16(\mathrm{a})$

Shubh Ashish
Shubh Ashish
Numerade Educator
00:25

Problem 11

Give the table for the group having the indicated digraph. In each digraph, take $e$ as identity element. List the identity $e$ first in your table, and list the remaining elements alphabetically, so that your answers will be easy to check.The digraph in Fig. $7.16(\mathrm{~b})$

Sam Limsuwannarot
Sam Limsuwannarot
Numerade Educator
02:42

Problem 12

Give the table for the group having the indicated digraph. In each digraph, take $e$ as identity element. List the identity $e$ first in your table, and list the remaining elements alphabetically, so that your answers will be easy to check.The digraph in Fig. $7.16(\mathrm{c})$

Shubh Ashish
Shubh Ashish
Numerade Educator
01:35

Problem 13

How can we tell from a Cayley digraph whether or not the corresponding group is commutative?

Nick Johnson
Nick Johnson
Numerade Educator
01:33

Problem 14

Using the condition found in Exercise 13 , show that the group corresponding to the Cayley digraph in Figure $7.13$ is not commutative.

James Kiss
James Kiss
Numerade Educator
05:10

Problem 15

Is it obvious from a Cayley digraph of a group whether or not the group is cyclic? [Hint: Look at Fig. 7.9(b).]

Ely Crowder
Ely Crowder
Numerade Educator
02:17

Problem 16

The large outside triangle in Fig. $7.11(\mathrm{~b})$ exhibits the cyclic subgroup $\{0,2,4\}$ of $\mathbb{Z}_{6}$. Does the smaller inside triangle similarly exhibit a cyclic subgroup of $\mathbb{Z}_{6}$ ? Why or why not?

Jay Patel
Jay Patel
Numerade Educator
01:56

Problem 17

The generating set $S=\{1,2\}$ for $\mathbb{E}_{6}$ contains more generators than necessary, since 1 is a generator for the group. Nevertheless, we can draw a Cayley digraph for $\mathbb{Z}_{6}$ with this generating set $S .$ Draw such a Cayley digraph.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:37

Problem 18

Draw a Cayley digraph for $\mathbb{Z}_{8}$ with generating set $S=\{2,5\}$.

AG
Ankit Gupta
Numerade Educator
01:38

Problem 19

A relation on a set $S$ of generators of a group $G$ is an equation that equates some product of generators and their inverses to the identity $e$ of $G$. For example, if $S=\{a, b\}$ and $G$ is commutative so that $a b=b a$, then one relation is $a b a^{-1} b^{-1}=e$. If, moreover, $b$ is its own inverse, then another relation is $b^{2}=e$.
a. Explain how we can find some relations on $S$ from a Cayley digraph of $G .$
b. Find three relations on the set $S=\{a, b]$ of generators for the group described by Fig. $7.13(\mathrm{~b})$.

James Chok
James Chok
Numerade Educator
03:01

Problem 20

Draw digraphs of the two possible structurally different groups of order 4, taking as small a generating set as possible in each case. You need not label vertices.

Chris Trentman
Chris Trentman
Numerade Educator
01:57

Problem 21

Use Cayley digraphs to show that for $n \geq 3$, there exists a nonabelian group with $2 n$ elements that is generated by two elements of order 2 .

Wendi Zhao
Wendi Zhao
Numerade Educator
01:57

Problem 22

Prove that there are at least three different abelian groups of order 8. [Hint: Find a Cayley digraph for a group of order 8 having one generator of order 4 and another of order $2 .$ Find a second Cayley digraph for a group of order 8 having three generators each with order $2 .]$.

Wendi Zhao
Wendi Zhao
Numerade Educator