Chapter Questions
Prove that the mapping given in Example 2 is a homomorphism.
Prove that the mapping given in Example 3 is a homomorphism.
Prove that the mapping given in Example 4 is a homomorphism.
Prove that the mapping given in Example 11 is a homomorphism.
Let $\mathbf{R}^{*}$ be the group of nonzero real numbers under multiplication, and let $r$ be a positive integer. Show that the mapping that takes $x$ to $x^{r}$ is a homomorphism from $\mathbf{R}^{*}$ to $\mathbf{R}^{*}$ and determine the kernel. Which values of $r$ yield an isomorphism?
Let $G$ be the group of all polynomials with real coefficients under addition. For each $f$ in $G$, let $\int f$ denote the antiderivative of $f$ that passes through the point $(0,0)$. Show that the mapping $f \rightarrow \int f$ from $G$ to $G$ is a homomorphism. What is the kernel of this mapping? Is this mapping a homomorphism if $\int f$ denotes the antiderivative of $f$ that passes through $(0,1)$ ?
If $\phi$ is a homomorphism from $G$ to $H$ and $\sigma$ is a homomorphism from $H$ to $K$, show that $\sigma \phi$ is a homomorphism from $G$ to $K$. How are Ker $\phi$ and Ker $\sigma \phi$ related? If $\phi$ and $\sigma$ are onto and $G$ is finite, describe [Ker $\sigma \phi: \operatorname{Ker} \phi]$ in terms of $|H|$ and $|K|$.
Let $G$ be a group of permutations. For each $\sigma$ in $G$, define$$\operatorname{sgn}(\sigma)=\left\{\begin{array}{ll}+1 & \text { if } \sigma \text { is an even permutation } \\-1 & \text { if } \sigma \text { is an odd permutation. }\end{array}\right.$$Prove that sgn is a homomorphism from $G$ to the multiplicative group $\{+1,-1\} .$ What is the kernel? Why does this homomorphism allow you to conclude that $A_{n}$ is a normal subgroup of $S_{n}$ of index 2 ? Why does this prove Exercise 23 of Chapter 5 ?
Prove that the mapping from $G \oplus H$ to $G$ given by $(g, h) \rightarrow g$ is a homomorphism. What is the kernel? This mapping is called the projection of $G \oplus H$ onto $G$.
Let $G$ be a subgroup of some dihedral group. For each $x$ in $G$, define$$\phi(x)=\left\{\begin{array}{ll}+1 & \text { if } x \text { is a rotation, } \\-1 & \text { if } x \text { is a reflection. }\end{array}\right.$$Prove that $\phi$ is a homomorphism from $G$ to the multiplicative group $\{+1,-1\}$. What is the kernel? Why does this prove Exercise 26 of Chapter 3?
Prove that $(Z \oplus Z) /(\langle(a, 0)\rangle \times\langle(0, b)\rangle)$ is isomorphic to $Z_{a} \oplus Z_{b}$.
Suppose that $k$ is a divisor of $n$. Prove that $Z_{n} /\langle k\rangle \approx Z_{k}$.
Prove that $(A \oplus B) /(A \oplus\{e\}) \approx B$.
Explain why the correspondence $x \rightarrow 3 x$ from $Z_{12}$ to $Z_{10}$ is not a homomorphism.
Suppose that $\phi$ is a homomorphism from $Z_{30}$ to $Z_{30}$ and Ker $\phi=$ $\{0,10,20\} .$ If $\phi(23)=9$, determine all elements that map to $9 .$
Prove that there is no homomorphism from $Z_{8} \oplus Z_{2}$ onto $Z_{4} \oplus Z_{4}$.
Prove that there is no homomorphism from $Z_{16} \oplus Z_{2}$ onto $Z_{4} \oplus Z_{4}$.
Can there be a homomorphism from $Z_{4} \oplus Z_{4}$ onto $Z_{8}$ ? Can there be a homomorphism from $Z_{16}$ onto $Z_{2} \oplus Z_{2} ?$ Explain your answers.
Suppose that there is a homomorphism $\phi$ from $Z_{17}$ to some group and that $\phi$ is not one-to-one. Determine $\phi$.
How many homomorphisms are there from $Z_{20}$ onto $Z_{8} ?$ How many are there to $Z_{8}$ ?
If $\phi$ is a homomorphism from $Z_{30}$ onto a group of order 5, determine the kernel of $\phi$.
Suppose that $\phi$ is a homomorphism from a finite group $G$ onto $\bar{G}$ and that $\bar{G}$ has an element of order 8 . Prove that $G$ has an element of order 8. Generalize.
Let $\phi$ be a homomorphism from a finite group $G$ to $\bar{G}$. If $H$ is a subgroup of $\bar{G}$ give a formula for $\left|\phi^{-1}(H)\right|$ in terms of $|H|$ and $\phi$.
Suppose that $\phi: Z_{50} \rightarrow Z_{15}$ is a group homomorphism with $\phi(7)=6$.a. Determine $\phi(x)$.b. Determine the image of $\phi$.c. Determine the kernel of $\phi$.d. Determine $\phi^{-1}(3) .$ That is, determine the set of all elements that map to 3 .
How many homomorphisms are there from $Z_{20}$ onto $Z_{10}$ ? How many are there to $Z_{10}$ ?
Determine all homomorphisms from $Z_{4}$ to $Z_{2} \oplus Z_{2}$.
Determine all homomorphisms from $Z_{n}$ to itself.
Suppose that $\phi$ is a homomorphism from $S_{4}$ onto $Z_{2} .$ Determine Ker $\phi$. Determine all homomorphisms from $S_{4}$ to $Z_{2}$.
Suppose that there is a homomorphism from a finite group $G$ onto $Z_{10}$. Prove that $G$ has normal subgroups of indexes 2 and 5 .
Suppose that $\phi$ is a homomorphism from a group $G$ onto $Z_{6} \oplus Z_{2}$ and that the kernel of $\phi$ has order $5 .$ Explain why $G$ must have normal subgroups of orders $5,10,15,20,30$, and 60 .
Suppose that $\phi$ is a homomorphism from $U(30)$ to $U(30)$ and that Ker $\phi=\{1,11\} .$ If $\phi(7)=7$, find all elements of $U(30)$ that map to 7 .
Find a homomorphism $\phi$ from $U(30)$ to $U(30)$ with kernel $\{1,11\}$ and $\phi(7)=7$.
Suppose that $\phi$ is a homomorphism from $U(40)$ to $U(40)$ and that Ker $\phi=\{1,9,17,33\} .$ If $\phi(11)=11$, find all elements of $U(40)$ that map to 11 .
Prove that there is no homomorphism from $A_{4}$ onto $Z_{2}$.
Prove that the mapping $\phi: Z \oplus Z \rightarrow Z$ given by $(a, b) \rightarrow a-b$ is a homomorphism. What is the kernel of $\phi ?$ Describe the set $\phi^{-1}(3)$ (that is, all elements that map to 3 ).
Suppose that there is a homomorphism $\phi$ from $Z \oplus Z$ to a group $G$ such that $\phi((3,2))=a$ and $\phi((2,1))=b$. Determine $\phi((4,4))$ in terms of $a$ and $b$. Assume that the operation of $G$ is addition.
Let $H=\left\{z \in C^{*}|| z \mid=1\right\}$. Prove that $C^{*} / H$ is isomorphic to $\mathbf{R}^{+}$, the group of positive real numbers under multiplication. $\left(\right.$ Recall $\left.|a+b i|=\sqrt{a^{2}+b^{2}} .\right)$
Let $\alpha$ be a homomorphism from $G_{1}$ to $H_{1}$ and $\beta$ be a homomorphism from $G_{2}$ to $H_{2}$. Determine the kernel of the homomorphism $\gamma$ from $G_{1} \oplus G_{2}$ to $H_{1} \oplus H_{2}$ defined by $\gamma\left(g_{1}, g_{2}\right)=\left(\alpha\left(g_{1}\right), \beta\left(g_{2}\right)\right)$.
Prove that the mapping $x \rightarrow x^{6}$ from $\mathbf{C}^{*}$ to $\mathbf{C}^{*}$ is a homomorphism. What is the kernel?
For each pair of positive integers $m$ and $n$, we can define a homomorphism from $Z$ to $Z_{m} \oplus Z_{n}$ by $x \rightarrow(x \bmod m, x \bmod n) .$ What is the kernel when $(m, n)=(3,4) ?$ What is the kernel when $(m, n)=$ $(6,4) ?$ Generalize.
(Second Isomorphism Theorem) If $K$ is a subgroup of $G$ and $N$ is a normal subgroup of $G$, prove that $K /(K \cap N)$ is isomorphic to $\mathrm{KN} / \mathrm{N}$.
(Third Isomorphism Theorem) If $M$ and $N$ are normal subgroups of $G$ and $N \leq M$, prove that $(G / N) /(M / N) \approx G / M .$ Think of this as a form of "cancelling out" the $N$ in the numerator and denominator.
Prove that the only homomorphism from $A_{4}$ to a finite group with order not divisible by 3 is the trivial mapping that takes every element to the identity.
Let $k$ be a divisor of $n .$ Consider the homomorphism from $U(n)$ to $U(k)$ given by $x \rightarrow x \bmod k$. What is the relationship between this homomorphism and the subgroup $U_{k}(n)$ of $U(n) ?$
Determine all homomorphic images of $D_{4}$ (up to isomorphism).
Let $N$ be a normal subgroup of a finite group $G .$ Use the theorems of this chapter to prove that the order of the group element $g N$ in $G / N$ divides the order of $g$.
Suppose that $G$ is a finite group and that $Z_{10}$ is a homomorphic image of $G$. What can we say about $|G|$ ? Generalize.
Suppose that $Z_{10}$ and $Z_{15}$ are both homomorphic images of a finite group $G$. What can be said about $|G|$ ? Generalize.
Suppose that for each prime $p, Z_{p}$ is the homomorphic image of a group $G .$ What can we say about $|G| ?$ Give an example of such a group.
(For students who have had linear algebra.) Suppose that $x$ is a particular solution to a system of linear equations and that $S$ is the entire solution set of the corresponding homogeneous system of linear equations. Explain why property 6 of Theorem $10.1$ guarantees that $x+S$ is the entire solution set of the nonhomogeneous system. In particular, describe the relevant groups and the homomorphism between them.
Let $N$ be a normal subgroup of a group $G .$ Use property 7 of Theorem $10.2$ to prove that every subgroup of $G / N$ has the form $H / N$, where $H$ is a subgroup of $G .$ (This exercise is referred to in Chapter 11 and Chapter $24 .$ )
Show that a homomorphism defined on a cyclic group is completely determined by its action on a generator of the group.
Use the First Isomorphism Theorem to prove Theorem $9.4 .$
Determine all homomorphisms from $D_{5}$ onto $Z_{2} \oplus Z_{2} .$ Determine all homomorphisms from $D_{5}$ to $Z_{2} \oplus Z_{2}$.
Let $Z[x]$ be the group of polynomials in $x$ with integer coefficients under addition. Prove that the mapping from $Z[x]$ into $Z$ given by $f(x) \rightarrow f(3)$ is a homomorphism. Give a geometric description of the kernel of this homomorphism. Generalize.
Prove that the mapping from $\mathbf{R}$ under addition to $S L(2, \mathbf{R})$ that takes $x$ to$$\left[\begin{array}{rr}\cos x & \sin x \\-\sin x & \cos x\end{array}\right]$$is a group homomorphism. What is the kernel of the homomorphism?
Suppose there is a homomorphism $\phi$ from $G$ onto $Z_{2} \oplus Z_{2} .$ Prove that $G$ is the union of three proper normal subgroups.
If $H$ and $K$ are normal subgroups of $G$ and $H \cap K=\{e\}$, prove that $G$ is isomorphic to a subgroup of $G / H \oplus G / K$.
If $\phi$ is a homomorphism from $G$ onto $H$, prove that $\phi(Z(G)) \subseteq Z(H)$.
Suppose that $\phi$ is a homomorphism from $D_{12}$ onto $D_{3} .$ What is $\phi\left(R_{180}\right) ?$
Prove that every group of order 77 is cyclic.
Determine all homomorphisms from $Z$ onto $S_{3} .$ Determine all homomorphisms from $Z$ to $S_{3}$.
Let $G$ be an Abelian group. Determine all homomorphisms from $S_{3}$ to $G$.
If $m$ and $n$ are positive integers prove that the mapping from $Z_{m}$ to $Z_{n}$ given by $\phi(x)=x \bmod n$ is a homomorphism if and only if $n$ divides $m$.
Prove that the mapping from $\mathbf{C}^{*}$ to $\mathbf{C}^{*}$ given by $\phi(x)=x^{2}$ is a homomorphism and that $\mathbf{C}^{*} /\{1,-1\}$ is isomorphic to $\mathbf{C}^{*}$. What happens if $\mathbf{C}^{*}$ is replaced by $\mathbf{R}^{*}$ ?
Let $p$ be a prime. Determine the number of homomorphisms from $Z_{p} \oplus Z_{p}$ into $Z_{p}$.