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Contemporary Abstract Algebra

Joseph Gallian

Chapter 6

Isomorphisms - all with Video Answers

Educators

Chapter Questions

01:52

Problem 1

Find an isomorphism from the group of integers under addition to the group of even integers under addition.

WM
William Mead
Numerade Educator
07:20

Problem 2

Find $\operatorname{Aut}(Z)$.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:51

Problem 3

Let $\mathbf{R}^{+}$ be the group of positive real numbers under multiplication. Show that the mapping $\phi(x)=\sqrt{x}$ is an automorphism of $\mathbf{R}^{+}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:58

Problem 4

Show that $U(8)$ is not isomorphic to $U(10)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:58

Problem 5

Show that $U(8)$ is isomorphic to $U(12)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
16:59

Problem 6

Prove that isomorphism is an equivalence relation. That is, for any groups $G, H$, and $K$ $G \approx G$
$G \approx H$ implies $H \approx G$
$G \approx H$ and $H \approx K$ implies $G \approx K$.

Chris Trentman
Chris Trentman
Numerade Educator
01:05

Problem 7

Prove that $S_{4}$ is not isomorphic to $D_{12}$.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:02

Problem 8

Show that the mapping $a \rightarrow \log _{10} a$ is an isomorphism from $\mathbf{R}^{+}$ under multiplication to $\mathbf{R}$ under addition.

WM
William Mead
Numerade Educator
10:08

Problem 9

In the notation of Theorem $6.1$, prove that $T_{e}$ is the identity and that $\left(T_{g}\right)^{-1}=T_{g^{-1}}$.

ET
Ed Tam
Numerade Educator
02:05

Problem 10

Given that $\phi$ is a isomorphism from a group $G$ under addition to a group $\bar{G}$ under addition, convert property 2 of Theorem $6.2$ to additive notation.

Wendi Zhao
Wendi Zhao
Numerade Educator
View

Problem 11

Let $G$ be a group under multiplication, $\bar{G}$ be a group under addition and $\phi$ be an isomorphism from $\mathrm{G}$ to $\bar{G}$. If $\phi(a)=\bar{a}$ and $\phi(b)=\bar{b}$, find an expression for $\phi\left(a^{3} b^{-2}\right)$ in terms of $\bar{a}$ and $\bar{b}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:51

Problem 12

Let $G$ be a group. Prove that the mapping $\alpha(g)=g^{-1}$ for all $g$ in $G$ is an automorphism if and only if $G$ is Abelian.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 13

If $g$ and $h$ are elements from a group, prove that $\phi_{g} \phi_{h}=\phi_{g h} .$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:28

Problem 14

Find two groups $G$ and $H$ such that $G \neq H$, but $\operatorname{Aut}(G) \approx \operatorname{Aut}(H)$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:38

Problem 15

Prove the assertion in Example 14 that the inner automorphisms $\phi_{R_{0}}, \phi_{R_{90}}, \phi_{H}$, and $\phi_{D}$ of $D_{4}$ are distinct.

Geno Ellis
Geno Ellis
Numerade Educator
07:20

Problem 16

Find $\operatorname{Aut}\left(Z_{6}\right)$.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:58

Problem 17

If $G$ is a group, prove that $\operatorname{Aut}(G)$ and $\operatorname{Inn}(G)$ are groups. (This exercise is referred to in this chapter.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 18

If a group $G$ is isomorphic to $H$, prove that $\operatorname{Aut}(G)$ is isomorphic to $\operatorname{Aut}(H)$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:01

Problem 19

Suppose $\phi$ belongs to $\operatorname{Aut}\left(Z_{n}\right)$ and $a$ is relatively prime to $n .$ If $\phi(a)=b$, determine a formula for $\phi(x)$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
06:17

Problem 20

Let $H$ be the subgroup of all rotations in $D_{n}$ and let $\phi$ be an automorphism of $D_{n} .$ Prove that $\phi(H)=H .$ (In words, an automorphism of $D_{n}$ carries rotations to rotations.)

Jenny Wu
Jenny Wu
Numerade Educator
01:05

Problem 21

Let $H=\left\{\beta \in S_{5} \mid \beta(1)=1\right\}$ and $K=\left\{\beta \in S_{5} \mid \beta(2)=2\right\} .$ Prove
that $H$ is isomorphic to $K .$ Is the same true if $S_{5}$ is replaced by $S_{n}$, where $n \geq 3 ?$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:40

Problem 22

Show that $Z$ has infinitely many subgroups isomorphic to $Z$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:34

Problem 23

Let $n$ be an even integer greater than 2 and let $\phi$ be an automorphism of $D_{n}$. Determine $\phi\left(R_{180}\right)$.

Clarissa Noh
Clarissa Noh
Numerade Educator
06:17

Problem 24

Let $\phi$ be an automorphism of a group $G$. Prove that $H=\{x \in G \mid$ $\phi(x)=x\}$ is a subgroup of $G$

Jenny Wu
Jenny Wu
Numerade Educator
06:02

Problem 25

Give an example of a cyclic group of smallest order that contains both a subgroup isomorphic to $Z_{12}$ and a subgroup isomorphic to $Z_{20} .$ No need to prove anything, but explain your reasoning.

Ely Crowder
Ely Crowder
Numerade Educator
01:57

Problem 26

Suppose that $\phi: Z_{20} \rightarrow Z_{20}$ is an automorphism and $\phi(5)=5$. What are the possibilities for $\phi(x)$ ?

Sherrie Fenner
Sherrie Fenner
Numerade Educator
01:40

Problem 27

Identify a group $G$ that has subgroups isomorphic to $Z_{n}$ for all positive integers $n$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:35

Problem 28

Prove that the mapping from $U(16)$ to itself given by $x \rightarrow x^{3}$ is an automorphism.

Anthony Ramos
Anthony Ramos
Numerade Educator
06:45

Problem 29

Let $r \in U(n)$. Prove that the mapping $\alpha: Z_{n} \rightarrow Z_{n}$ defined by $\alpha(s)=$ $s r \bmod n$ for all $s$ in $Z_{n}$ is an automorphism of $Z_{n}$. (This exercise is referred to in this chapter.)

Sandip Ranjan
Sandip Ranjan
Numerade Educator
03:53

Problem 30

The group $\left\{\left[\begin{array}{ll}1 & a \\ 0 & 1\end{array}\right] \mid a \in Z\right\}$ is isomorphic to what familiar group? What if $Z$ is replaced by $\mathbf{R}$ ?

Anthony Ramos
Anthony Ramos
Numerade Educator
02:19

Problem 31

If $\phi$ and $\gamma$ are isomorphisms from the cyclic group $\langle a\rangle$ to some group and $\phi(a)=\gamma(a)$, prove that $\phi=\gamma$.

Narayan Hari
Narayan Hari
Numerade Educator
04:16

Problem 32

Suppose that $\phi: Z_{50} \rightarrow Z_{50}$ is an automorphism with $\phi(7)=13$. Determine a formula for $\phi(x)$.

Dillon Huddleston
Dillon Huddleston
Numerade Educator
01:06

Problem 33

Prove property 1 of Theorem $6.3$.

Carson Merrill
Carson Merrill
Numerade Educator
01:32

Problem 34

Prove property 4 of Theorem $6.3 .$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
01:02

Problem 35

Referring to Theorem $6.1$, prove that $T_{g}$ is indeed a permutation on the set $G$.

Teresa Fuston
Teresa Fuston
Numerade Educator
03:58

Problem 36

Prove or disprove that $U(20)$ and $U(24)$ are isomorphic.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:33

Problem 37

Show that the mapping $\phi(a+b i)=a-b i$ is an automorphism of the group of complex numbers under addition. Show that $\phi$ preserves complex multiplication as well-that is, $\phi(x y)=\phi(x) \phi(y)$ for all $x$ and $y$ in $\mathbf{C}$. (This exercise is referred to in Chapter 15.)

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:51

Problem 38

Let
$$G=\{a+b \sqrt{2} \mid a, b \text { are rational }\}$$
and
$$H=\left\{\left[\begin{array}{cc}
a & 2 b \\
b & a
\end{array}\right] \mid a, b \text { are rational }\right\}$$
Show that $G$ and $H$ are isomorphic under addition. Prove that $G$ and $H$ are closed under multiplication. Does your isomorphism preserve multiplication as well as addition? ( $G$ and $H$ are examples of rings a topic we will take up in Part $3 .$.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:58

Problem 39

Prove that $Z$ under addition is not isomorphic to $Q$ under addition.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:17

Problem 40

Explain why $S_{8}$ contains subgroups isomorphic to $Z_{15}, U(16)$, and $D_{8}$.

Lottie Adams
Lottie Adams
Numerade Educator
03:58

Problem 41

Let $\mathbf{C}$ be the complex numbers and
$$M=\left\{\left[\begin{array}{lr}
a & -b \\
b & a
\end{array}\right] \mid a, b \in \mathbf{R}\right\}$$
Prove that $\mathbf{C}$ and $M$ are isomorphic under addition and that $\mathbf{C}^{*}$ and $M^{*}$, the nonzero elements of $M$, are isomorphic under multiplication.

Anthony Ramos
Anthony Ramos
Numerade Educator
08:25

Problem 42

Let $\mathbf{R}^{n}=\left\{\left(a_{1}, a_{2}, \ldots, a_{n}\right) \mid a_{i} \in \mathbf{R}\right\} .$ Show that the mapping $\phi:\left(a_{1},\right.$,
$\left.a_{2}, \ldots, a_{n}\right) \rightarrow\left(-a_{1},-a_{2}, \ldots,-a_{n}\right)$ is an automorphism of the group $\mathbf{R}^{n}$ under componentwise addition. This automorphism is called inversion. Describe the action of $\phi$ geometrically.

Ely Crowder
Ely Crowder
Numerade Educator
01:57

Problem 43

Consider the following statement: The order of a subgroup divides the order of the group. Suppose you could prove this for finite permutation groups. Would the statement then be true for all finite groups? Explain.

Wendi Zhao
Wendi Zhao
Numerade Educator
View

Problem 44

Suppose that $G$ is a finite Abelian group and $G$ has no element of order 2 . Show that the mapping $g \rightarrow g^{2}$ is an automorphism of $G$. Show, by example, that there is an infinite Abelian group for which the mapping $g \rightarrow g^{2}$ is one-to-one and operation-preserving but not an automorphism.

Nick Johnson
Nick Johnson
Numerade Educator
06:17

Problem 45

Let $G$ be a group and let $g \in G .$ If $z \in Z(G)$, show that the inner automorphism induced by $g$ is the same as the inner automorphism induced by $z g$ (that is, that the mappings $\phi_{g}$ and $\phi_{z g}$ are equal).

Jenny Wu
Jenny Wu
Numerade Educator
01:05

Problem 46

Prove that $\mathbf{R}$ under addition is not isomorphic to $\mathbf{R}^{*}$ under multiplication.

Anthony Ramos
Anthony Ramos
Numerade Educator
06:17

Problem 47

Suppose that $g$ and $h$ induce the same inner automorphism of a group $G$. Prove that $h^{-1} g \in Z(G)$.

Jenny Wu
Jenny Wu
Numerade Educator
01:26

Problem 48

Combine the results of Exercises 45 and 47 into a single "if and only if" theorem.

Christopher Stanley
Christopher Stanley
Numerade Educator
09:50

Problem 49

If $\alpha$ and $\beta$ are elements in $S_{n}(n \geq 3)$,prove that $\phi_{\alpha}=\phi_{\beta}$ implies that $\alpha=\beta$. (Here, $\phi_{\alpha}$ is the inner automorphism of $S_{n}$ induces by $\alpha .$ )

Aayush Gupta
Aayush Gupta
Numerade Educator
View

Problem 50

Prove or disprove that the mapping $\phi$ from $Q^{+}$, the positive rational numbers under multiplication, to itself given by $\phi(x)=x^{2}$ is an automorphism.

Nick Johnson
Nick Johnson
Numerade Educator
06:17

Problem 51

Suppose the $\phi$ and $\gamma$ are isomorphisms of some group $G$ to the same group. Prove that $H=\{g \in G \mid \phi(g)=\gamma(g)\}$ is a subgroup of $G$.

Jenny Wu
Jenny Wu
Numerade Educator
01:40

Problem 52

Let $G$ be a group. Complete the following statement: $|\operatorname{Inn}(G)|=1$ if and only if_________.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:17

Problem 53

Suppose that $G$ is an Abelian group and $\phi$ is an automorphism of $G$. Prove that $H=\left\{x \in G \mid \phi(x)=x^{-1}\right\}$ is a subgroup of $G$.

Jenny Wu
Jenny Wu
Numerade Educator
00:37

Problem 54

Let $\phi$ be an automorphism of $D_{8} .$ What are the possibilities for $\phi\left(R_{45}\right) ?$

AG
Ankit Gupta
Numerade Educator
00:20

Problem 55

Let $\phi$ be an automorphism of $\mathbf{C}^{*}$, the group of nonzero complex numbers under multiplcation. Determine $\phi(-1) .$ Determine the possibilities for $\phi(i)$.

Heather Zimmers
Heather Zimmers
Numerade Educator
01:51

Problem 56

Let $G=\{0, \pm 2, \pm 4, \pm 6, \ldots\}$ and $H=\{0, \pm 3, \pm 6, \pm 9, \ldots\}$
Prove that $G$ and $H$ are isomorphic groups under addition by defining a mapping that has the required properties. Does your isomorphism preserve multiplication? Generalize to the case when $G=\langle m\rangle$ and $H=\langle n\rangle$, where $m$ and $n$ are integers.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:20

Problem 57

Give three examples of groups of order 120, no two of which are isomophic. Explain why they are not isomorphic.

Ely Crowder
Ely Crowder
Numerade Educator
06:17

Problem 58

Let $\phi$ be an automorphism of $D_{4}$ such that $\phi(H)=D$. Find $\phi(V)$.

Jenny Wu
Jenny Wu
Numerade Educator
02:31

Problem 59

Suppose that $\phi$ is an automorphism of $D_{4}$ such that $\phi\left(R_{90}\right)=R_{270}$ and $\phi(V)=V$. Determine $\phi(D)$ and $\phi(H)$.

Jay Patel
Jay Patel
Numerade Educator
03:19

Problem 60

In $\operatorname{Aut}\left(Z_{9}\right)$, let $\alpha_{i}$ denote the automorphism that sends 1 to $i$ where $\operatorname{gcd}(i, 9)=1 .$ Write $\alpha_{5}$ and $\alpha_{8}$ as permutations of $\{0,1, \ldots, 8\}$ in disjoint cycle form. [For example, $\left.\alpha_{2}=(0)(124875)(36) .\right]$

Alexandra Embry
Alexandra Embry
Numerade Educator
00:24

Problem 61

Write the permutation corresponding to $R_{90}$ in the left regular representation of $D_{4}$ in cycle form.

AG
Ankit Gupta
Numerade Educator
01:15

Problem 62

Show that every automorphism $\phi$ of the rational numbers $Q$ under addition to itself has the form $\phi(x)=x \phi(1)$.

Manisha Sarker
Manisha Sarker
Numerade Educator
06:02

Problem 63

Prove that $Q^{+}$, the group of positive rational numbers under multiplication, is isomorphic to a proper subgroup of itself.

Ely Crowder
Ely Crowder
Numerade Educator
06:02

Problem 64

Prove that $Q$, the group of rational numbers under addition, is not isomorphic to a proper subgroup of itself.

Ely Crowder
Ely Crowder
Numerade Educator
08:25

Problem 65

Prove that every automorphism of $\mathbf{R}^{*}$, the group of nonzero real numbers under multiplication, maps positive numbers to positive numbers and negative numbers to negative numbers.

Ely Crowder
Ely Crowder
Numerade Educator
30:00

Problem 66

Prove that $Q^{*}$, the group of nonzero rational numbers under multiplication, is not isomorphic to $Q$, the group of rational numbers under addition.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
06:02

Problem 67

Give a group theoretic proof that $Q$ under addition is not isomorphic to $\mathbf{R}^{+}$ under multiplication.

Ely Crowder
Ely Crowder
Numerade Educator