Group Homomorphisms

Prove that the mapping given in Example 2 is a homomorphism.

Example 2: Let $\mathbb {R}^*$ be the group of nonzero real numbers under multiplication. Then the determinant mapping $A \to \det A$ is a homomorphism from $GL(2, \mathbb {R})$ to $\mathbb {R}^*$. The kernel of the determinant mapping is $SL(2, \mathbb {R})$.

Prove that the mapping given in Example 3 is a homomorphism.

Example 3: The mapping $\phi$ from $\mathbb {R}^*$ to $\mathbb {R}^*$, defined by $\phi (x) = \left|x\right|$, is a homomorphism with $\operatorname {Ker} \phi = \left\{ 1,-1\right\}$.

Prove that the mapping given in Example 4 is a homomorphism.

Example 4: Let $\mathbb {R}^{x}$ denote the group of all polynomials with real coefficients under addition. For any $f$ in $\mathbb {R}^{x}$, let $f'$ denote the derivative of $f$. Then the mapping $f \to f'$ is a homomorphism from $\mathbb {R}^{x}$ to itself. The kernel of the derivative mapping is the set of all constant polynomials.

Prove that the mapping given in Example 12 is a homomorphism.

Example 12: The mapping from $S_n$ to $\mathbb {Z}_2$ that takes an even permutation to 0 and an odd permutation to 1 is a homomorphism.

Let $\mathbb {R}^*$ 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 $\mathbb {R}^*$ to $\mathbb {R}^*$ 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 $f \to \int f$ 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 $\operatorname {Ker} \phi$ and $\operatorname {Ker} \sigma \phi$ related? If $\phi$ and $\sigma$ are onto and $G$ is finite, describe $\left[\operatorname {Ker} \sigma \phi : \operatorname {Ker} \phi \right]$ in terms of $\left|H\right|$ and $\left|K\right|$.

Let $G$ be a group of permutations. For each $\sigma$ in $G$, define

Prove that $\operatorname {sgn}$ is a homomorphism from $G$ to the multiplicative group $\left\{ +1,-1\right\}$. 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 27 of Chapter 5?

Prove that the mapping from $G \oplus H$ to $G$ given by $(g,h) \to g$ 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

Prove that $\phi$ is a homomorphism from $G$ to the multiplicative group $\left\{ +1,-1\right\}$. What is the kernel? Why does this prove Exercise 30 of Chapter 3?

Prove that $(\mathbb {Z} \oplus \mathbb {Z})/( \left\langle (a,0)\right\rangle \oplus \left\langle (0,b)\right\rangle )$ is isomorphic to $\mathbb {Z}_a \oplus \mathbb {Z}_b$.

Suppose that $k$ is a divisor of $n$. Prove that $\mathbb {Z}_n/ \left\langle k\right\rangle \approx \mathbb {Z}_k$.

Prove that $(A \oplus B)/(A \oplus \left\{ e\right\} ) \approx B$.

Explain why the correspondence $x \to 3x$ from $\mathbb {Z}_{12}$ to $\mathbb {Z}_{10}$ is not a homomorphism.

Suppose that $\phi$ is a homomorphism from $\mathbb {Z}_{30}$ to $\mathbb {Z}_{30}$ and $\operatorname {Ker}\phi = \left\{ 0,10,20\right\}$. If $\phi (23) = 9$, determine all elements that map to 9.

Prove that there is no homomorphism from $\mathbb {Z}_8 \oplus \mathbb {Z}_2$ onto $\mathbb {Z}_4 \oplus \mathbb {Z}_4$.

Prove that there is no homomorphism from $\mathbb {Z}_{16} \oplus \mathbb {Z}_2$ onto $\mathbb {Z}_4 \oplus \mathbb {Z}_4$.

Can there be a homomorphism from $\mathbb {Z}_4 \oplus \mathbb {Z}_4$ onto $\mathbb {Z}_8$? Can there be a homomorphism from $\mathbb {Z}_{16}$ onto $\mathbb {Z}_2 \oplus \mathbb {Z}_2$? Can there be a homomorphism from $\mathbb {Z}_{18}$ onto $\mathbb {Z}_3 \oplus \mathbb {Z}_2$? Explain your answers.

Suppose that there is a homomorphism $\phi$ from $\mathbb {Z}_{17}$ to some group and that $\phi$ is not one-to-one. Determine $\phi$.

How many homomorphisms are there from $\mathbb {Z}_{20}$ onto $\mathbb {Z}_8$? How many are there to $\mathbb {Z}_8$?

If $\phi$ is a homomorphism from $\mathbb {Z}_{30}$ onto a group of order 5, determine the kernel of $\phi$.

Suppose that $\phi$ is a homomorphism from a finite group $G$ onto $\overline{G}$ and that $\overline{G}$ 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 $\overline{G}$. If $H$ is a subgroup of $\overline{G}$ give a formula for $\left|\phi^{-1}(H)\right|$ in terms of $\left|H\right|$ and $\phi$.

Suppose that $\phi : \mathbb {Z}_{50} \to \mathbb {Z}_{15}$ is a group homomorphism with $\phi (7) = 6$.

How many homomorphisms are there from $\mathbb {Z}_{20}$ onto $\mathbb {Z}_{10}$? How many are there to $\mathbb {Z}_{10}$?

Determine all homomorphisms from $\mathbb {Z}_4$ to $\mathbb {Z}_2 \oplus \mathbb {Z}_2$.

Determine all homomorphisms from $\mathbb {Z}_n$ to itself.

Suppose that $\phi$ is a homomorphism from $S_4$ onto $\mathbb {Z}_2$. Determine $\operatorname {Ker}\phi$. Determine all homomorphisms from $S_4$ to $\mathbb {Z}_2$.

Suppose that there is a homomorphism from a finite group $G$ onto $\mathbb {Z}_{10}$. Prove that $G$ has normal subgroups of indexes 2 and 5.

Suppose that $\phi$ is a homomorphism from a group $G$ onto $\mathbb {Z}_6 \oplus \mathbb {Z}_2$ 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 $\operatorname {Ker}\phi = \left\{ 1,11\right\}$. If $\phi (7) = 7$, find all elements of $U(30)$ that map to 7.

Suppose that $\phi$ is a homomorphism from $\mathbb {Z}_{30}$ to $\mathbb {Z}_{30}$ and that $\operatorname {Ker}\phi = \left\{ 0,5,10,15,20,25\right\}$. If $\phi (7) = 12$, find all elements of $\mathbb {Z}_{30}$ that map to 24. Find all elements of $\mathbb {Z}_{30}$ that map to 6.

Suppose that $\phi$ is a homomorphism from $U(40)$ to $U(40)$ and that $\operatorname {Ker}\phi = \left\{ 1,9,17,33\right\}$. If $\phi (11) = 11$, find all elements of $U(40)$ that map to 11.

If $\phi$ is a homomorphism from $A_4$ onto a cyclic group $G$, prove that $\left|G\right| = 1$ or 3.

Prove that the mapping $\phi : \mathbb {Z} \oplus \mathbb {Z} \to \mathbb {Z}$ given by $(a,b) \to a-b$ is a homomorphism. What is the kernel of $\phi$? Describe the set $\phi^{-1}(3)$ (i.e., all elements that map to 3).

Suppose that there is a homomorphism $\phi$ from $\mathbb {Z} \oplus \mathbb {Z}$ 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 $H = \left\{ z \in \mathbb {C}^* \mid \left|z\right| = 1\right\}$. Prove that $\mathbb {C}^*/H$ is isomorphic to $\mathbb {R}^+$, the group of positive real numbers under multiplication. (Recall $\left|a+bi\right| = \sqrt{a^2+b^2}$.)

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 $\gamma (g_1, g_2) = (\alpha (g_1), \beta (g_2))$.

Prove that the mapping $x \to x^6$ from $\mathbb {C}^*$ to $\mathbb {C}^*$ is a homomorphism. What is the kernel?

For each pair of positive integers $m$ and $n$, we can define a homomorphism from $\mathbb {Z}$ to $\mathbb {Z}_m \oplus \mathbb {Z}_n$ by $x \to (x \bmod m, x \bmod n)$. What is the kernel when $(m,n) = (3,4)$? What is the kernel when $(m,n) = (6,4)$? Generalize.

If $G$ is an Abelian group and $\phi$ is a homomorphism from $G$ to some group, prove that $H = \left\{ x \in G \mid \phi (x) = x^{-2}\right\}$ is a subgroup of $G$. If $\phi (x) = x^3$, what can you say about the orders of the elements of $H$?

If $\phi$ is a homomorphism from $D_6$ onto $D_3$, what is $\operatorname {Ker}\phi$?

Prove that there is no homomorphism from $\mathbb {Z}_{16}$, $\mathbb {Z}_8 \oplus \mathbb {Z}_2$ or $\mathbb {Z}_4 \oplus \mathbb {Z}_4$ onto $\mathbb {Z}_2 \oplus \mathbb {Z}_2 \oplus \mathbb {Z}_2$. Generalize to $\mathbb {Z}_{2^m}$ and $\mathbb {Z}_{2^m} \oplus \mathbb {Z}_{2^n}$ onto $\mathbb {Z}_2 \oplus \mathbb {Z}_2 \oplus \mathbb {Z}_2$. Does your argument work for $\mathbb {Z}_{p^m}$ and $\mathbb {Z}_{p^m} \oplus \mathbb {Z}_{p^n}$ onto $\mathbb {Z}_p \oplus \mathbb {Z}_p \oplus \mathbb {Z}_p$ where $p$ is any prime?

Prove that is no homomorphism from $S_3$ onto $\mathbb {Z}_3$. Prove that there is no homomorphism from $S_4$ onto $\mathbb {Z}_3$.

Use the subgroup $H$ defined in Example 9 in Chapter 9 to prove that there is a homomorphism from $S_4$ onto $S_3$. This exercise is referred to in this chapter.

Example 9: Referring to the group table for $A_4$, $H = \left\{ \alpha_1, \alpha_2, \alpha_3, \alpha_4\right\}$ is a normal subgroup of $A_4$ of order 4 (and, in fact, is normal in $S_4$).

Prove that every homomorphic image of $\mathbb {Z}_m \oplus \mathbb {Z}_n$ has the form $\mathbb {Z}_s \oplus \mathbb {Z}_t$ where $s$ divides $m$ and $t$ divides $n$.

Let $s$ and $t$ be relatively prime positive integers. Prove that $U(st)/U_s(st)$ is isomorphic to $U(s)$.

If $G = \left\langle S\right\rangle$ and $\phi$ is a homomorphism from $G$ to some group, prove that $\phi (G) = \left\langle \phi (S)\right\rangle$.

If $K$ is a subgroup of $G$ and $N$ is a normal subgroup of $G$, prove that $K/(K \cap N)$ is isomorphic to $KN/N$.

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 $x \to x \bmod k$. 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 $gN$ in $G/N$ divides the order of $g$.

Suppose that $G$ is a finite group and that $\mathbb {Z}_{10}$ is a homomorphic image of $G$. What can we say about $\left|G\right|$? Generalize.

Suppose that $\mathbb {Z}_{10}$ and $\mathbb {Z}_{15}$ are both homomorphic images of a finite group $G$. What can be said about $\left|G\right|$? Generalize.

Suppose that for each prime $p$, $\mathbb {Z}_p$ is the homomorphic image of a group $G$. What can we say about $\left|G\right|$? 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 (if $\phi (g) = g'$, then $\phi^{-1}(g') = \left\{ x \in G \mid \phi (x) = g'\right\} = g\operatorname {Ker}\phi$) 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 8 of Theorem 10.2 (if $K$ is a subgroup of $\overline{G}$, then $\phi^{-1}(K) = \left\{ k \in G \mid \phi (k) \in K\right\}$ is a subgroup of $G$) 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 23.)

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.

Theorem 9.4: For any group $G$, $G/Z(G)$ is isomorphic to $\operatorname {Inn}(G)$.

Determine all homomorphisms from $D_5$ onto $\mathbb {Z}_2 \oplus \mathbb {Z}_2$. Determine all homomorphisms from $D_5$ to $\mathbb {Z}_2 \oplus \mathbb {Z}_2$.

Let $\mathbb {Z}^{x}$ be the group of polynomials in $x$ with integer coefficients under addition. Prove that the mapping from $\mathbb {Z}^{x}$ into $\mathbb {Z}$ given by $f(x) \to f(3)$ is a homomorphism. Give a geometric description of the kernel of this homomorphism. Generalize.

Prove that the mapping from $\mathbb {R}$ under addition to $SL(2, \mathbb {R})$ that takes $x$ to

is a group homomorphism. What is the kernel of the homomorphism?

Suppose there is a homomorphism $\phi$ from $G$ onto $\mathbb {Z}_2 \oplus \mathbb {Z}_2$. Prove that $G$ is the union of three proper normal subgroups.

If $H$ and $K$ are normal subgroups of $G$ and $H \cap K = \left\{ e\right\}$, prove that $G$ is isomorphic to a subgroup of $G/H \oplus G/K$.

Find a homomorphism from $\mathbb {Z}_8 \oplus \mathbb {Z}_4$ onto $\mathbb {Z}_4 \oplus \mathbb {Z}_4$.

Suppose $\phi$ is a homomorphism from $D_{12}$ onto $D_3$. What is $\phi (R_{180})$?

Does the argument given in Example 19 to prove every group of order 35 is cyclic remain valid for a group of order 55? If not, where does it fail?

Example 18 (the order-35 argument): Let $G$ be a group of order 35. By Lagrange's Theorem, every nonidentity element of $G$ has order 5, 7, or 35. If some element has order 35, $G$ is cyclic. So we may assume that all nonidentity elements have order 5 or 7. However, not all such elements can have order 5, since elements of order 5 come 4 at a time and 4 does not divide 34. Similarly, since 6 does not divide 34, not all nonidentity elements can have order 7. So, $G$ has elements of order 7 and order 5. Since $G$ has an element of order 7, it has a subgroup of order 7, call it $H$. In fact, $H$ is the only subgroup of $G$ of order 7, for if $K$ is another subgroup of $G$ of order 7, we have by Theorem 7.2 that $\left|HK\right| = \left|H\right|\left|K\right|/\left|H \cap K\right| = 7 \cdot 7/1 = 49$, which is impossible in a group of order 35. Since for every $a$ in $G$, $aHa^{-1}$ is also a subgroup of $G$ of order 7, we must have $aHa^{-1} = H$. So, $N(H) = G$. Since $H$ has prime order, it is cyclic and therefore Abelian, so $C(H)$ contains $H$. Thus 7 divides $\left|C(H)\right|$ and $\left|C(H)\right|$ divides 35, so $C(H) = G$ or $C(H) = H$. If $C(H) = G$, then we may obtain an element $x$ of order 35 by letting $x = hk$, where $h$ is a nonidentity element of $H$ and $k$ has order 5. On the other hand, if $C(H) = H$, then $\left|C(H)\right| = 7$ and $\left|N(H)/C(H)\right| = 35/7 = 5$. However, 5 does not divide $\left|\operatorname {Aut}(H)\right| = \left|\operatorname {Aut}(\mathbb {Z}_7)\right| = 6$. This contradiction shows that $G$ is cyclic.

If $G$ is a non-Abelian group and $\left|G\right| = 55$, prove that $G$ has exactly 11 subgroups of order 5 and they have the form $a^i K a^{-i}$ for $i = 0, 1, 2, \ldots , 10$ for some element $a$ in $G$ and some subgroup $K$ of $G$.

Prove that for any two primes $p$ and $q$ with $p < q$ where $p$ does not divide $q-1$, a group of order $pq$ is cyclic.

Determine all homomorphisms from $\mathbb {Z}$ onto $S_3$. Determine all homomorphisms from $\mathbb {Z}$ 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 $\mathbb {Z}_m$ to $\mathbb {Z}_n$ given by $\phi (x) = x \bmod n$ is a homomorphism if and only if $n$ divides $m$.

Prove that the mapping from $\mathbb {C}^*$ to $\mathbb {C}^*$ given by $\phi (x) = x^2$ is a homomorphism and that $\mathbb {C}^*/\left\{ 1,-1\right\}$ is isomorphic to $\mathbb {C}^*$. What happens if $\mathbb {C}^*$ is replaced by $\mathbb {R}^*$?

Let $p$ be a prime. Determine the number of homomorphisms from $\mathbb {Z}_p \oplus \mathbb {Z}_p$ into $\mathbb {Z}_p$.