Contemporary Abstract Algebra

Joseph Gallian

Chapter 6

Isomorphisms - all with Video Answers

Educators

Chapter Questions

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

Problem 2

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

Mengchun Cai

Numerade Educator

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

Problem 4

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

Anthony Ramos

Numerade Educator

Problem 5

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

Anthony Ramos

Numerade Educator

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 $H \approx G$, and $G \approx H$ and $H \approx K$ implies $G \approx K .$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 7

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

Anthony Ramos

Numerade Educator

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

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

Problem 10

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

Problem 11

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

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 12

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

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 13

Prove the assertion in Example 12 that the inner automorphisms $\phi_{R_{0}}, \phi_{R_{90}}, \phi_{H}$, and $\phi_{D}$ of $D_{4}$ are distinct.

Anthony Ramos

Numerade Educator

Problem 14

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

Mengchun Cai

Numerade Educator

Problem 15

If $G$ is a group, prove that $\operatorname{Aut}(G)$ and $\operatorname{Inn}(G)$ are groups.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 16

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

Problem 17

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

Numerade Educator

Problem 18

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

Numerade Educator

Problem 19

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

Numerade Educator

Problem 20

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

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 21

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

Julian Wong

Numerade Educator

Problem 22

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

Numerade Educator

Problem 23

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

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 24

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

Sherrie Fenner

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Problem 25

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

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 26

Prove that the mapping from $U(16)$ to itself given by $x \rightarrow x^{3}$ is an automorphism. What about $x \rightarrow x^{5}$ and $x \rightarrow x^{7} ?$ Generalize.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 27

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

Numerade Educator

Problem 28

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

Numerade Educator

Problem 29

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

Numerade Educator

Problem 30

Suppose that $\phi: Z_{50} \rightarrow Z_{50}$ is an automorphism with $\phi(11)=13$. Determine a formula for $\phi(x)$.

Dillon Huddleston

Dillon Huddleston

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Problem 31

Prove property 1 of Theorem $6.3$.

Carson Merrill

Numerade Educator

Problem 32

Prove property 4 of Theorem $6.3$.

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 33

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

Teresa Fuston

Numerade Educator

Problem 34

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

Anthony Ramos

Numerade Educator

Problem 35

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

Problem 36

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

Problem 37

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

Anthony Ramos

Numerade Educator

Problem 38

Prove that the quaternion group (see Exercise 4, Supplementary Exercises for Chapters $1-4$ ) is not isomorphic to the dihedral group $D_{4}$.

Ely Crowder

Numerade Educator

Problem 39

Let $\mathbf{C}$ be the complex numbers and
$$
M=\left\{\left[\begin{array}{rr}
a & -b \\
b & a
\end{array}\right] \mid a, b \in \mathbf{R}\right\}
$$
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

Numerade Educator

Problem 40

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 $\phi$ :
$\left(a_{1}, a_{2}, \ldots, a_{n}\right) \rightarrow\left(-a_{1},-a_{2}, \ldots,-a_{n}\right)$ 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

Numerade Educator

Problem 41

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

Numerade Educator

Problem 42

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

Numerade Educator

Problem 43

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

Numerade Educator

Problem 44

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

Problem 45

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

Jenny Wu

Numerade Educator

Problem 46

Combine the results of Exercises 43 and 45 into a single "if and only if" theorem.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 47

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

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 48

Let $\phi$ be an isomorphism from a group $G$ to a group $\bar{G}$ and let $a$ belong to $G$. Prove that $\phi(C(a))=C(\phi(a))$.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 49

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

Numerade Educator

Problem 50

Suppose that $\beta$ is an automorphism of a group $G$. Prove that $H=$ $\left\{g \in G \mid \beta^{2}(g)=g\right\}$ is a subgroup of $G$. Generalize.

Jenny Wu

Numerade Educator

Problem 51

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

Numerade Educator

Problem 52

Given a group $G$, define a new group $G^{*}$ that has the same elements as $G$ with the operation $*$ defines by $a * b=b a$ for all $a$ and $b$ in $G^{*}$. Prove that the mapping from $G$ to $G^{*}$ defined by $\phi(x)=x^{-1}$ for all $x$ in $G$ is an isomorphism from $G$ onto $G^{*}$.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 53

Let $a$ belong to a group $G$ and let $|a|$ be finite. Let $\phi_{a}$ be the automorphism of $G$ given by $\phi_{a}(x)=a x a^{-1}$. Show that $\left|\phi_{a}\right|$ divides $|a|$. Exhibit an element $a$ from a group for which $1<\left|\phi_{a}\right|<|a|$

Jenny Wu

Numerade Educator

Problem 54

Let $G=\{0, \pm 2, \pm 4, \pm 6, \ldots\}$ and $H=\{0, \pm 3, \pm 6, \pm 9, \ldots\}$.
Show that $G$ and $H$ are isomorphic groups under addition. 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

Problem 55

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

Numerade Educator

Problem 56

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

Problem 57

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

AG

Ankit Gupta

Numerade Educator

Problem 58

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

Manisha Sarker

Numerade Educator

Problem 59

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

Ely Crowder

Numerade Educator

Problem 60

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

Ely Crowder

Numerade Educator

Problem 61

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

Numerade Educator

Problem 62

Let $G$ be a finite group. Show that in the disjoint cycle form of the right regular representation $T_{g}(x)=x g$ of $G$, each cycle has length $|g|$.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 63

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

Ely Crowder

Numerade Educator