(Third Isomorphism Theorem) If M and N are normal

Problem 67E Suppose G is an Abelian group under addition with the property that for every positive integer n, the set nG ={ng | g ? G} = G. Show that every proper subgroup of G is properly contained in a proper subgroup of G. Name two familiar groups that satisfy the hypothesis.

Problem 1CE This software determines the homomorphisms from Zm to Zn. (Recall that a homomorphism from Zm is completely determined by the image of 1.) Run the program for m = 20 with various choices for n. Run the program for m = 15 with various choices for n. What relationship do you see between m and n and the number of homomorphisms from Zm to Zn? For each choice of m and n, observe the smallest positive image of 1. Try to see the relationship between this image and the values of m and n. What relationship do you see between the smallest positive image of 1 and the other images of 1? Test your conclusions with other choices of m and n.

Problem 1E Prove that the mapping given in Example 2 is a homomorphism. Reference: EXAMPLE 2 Let R* be the group of nonzero real numbers under multiplication. Then the determinant mapping A ? det A is a homomorphism from GL(2, R) to R*. The kernel of the determinant mapping is SL(2, R).

Problem 3E Prove that the mapping given in Example 4 is a homomorphism. Reference: EXAMPLE 4 Let R[x] denote the group of all polynomials with real coefficients under addition. For any f in R[x], let f ' denote the derivative of f. Then the mapping f ? f ' is a homomorphism from R[x] to itself. The kernel of the derivative mapping is the set of all constant polynomials.

Problem 2E Prove that the mapping given in Example 3 is a homomorphism. Reference: EXAMPLE 3 The mapping f from R* to R*, defined by ? (x) = |x|, is a homomorphism with Ker ? = {1, 21}.

Problem 4E Prove that the mapping given in Example 11 is a homomorphism. Reference: EXAMPLE 11 The mapping from Sn to Z2 that takes an even permutation to 0 and an odd permutation to 1 is a homomorphism. Figure 10.2 illustrates the telescoping nature of the mapping.

Problem 5E Let 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 xr is a homomorphism from R* to 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)?

Problem 7E If ? is a homomorphism from G to H and ? is a homomorphism from H to K, show that ?? is a homomorphism from G to K. How are Ker ? and Ker ?? related? If ? and ? are onto and G is finite, describe [Ker ??:Ker ?] in terms of |H| and |K|.

Problem 8E Let G be a group of permutations. For each ? in G, define 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 An is a normal subgroup of Sn of index 2? Why does this prove Exercise 23 of Chapter 5? Reference: Show that if H is a subgroup of Sn, then either every member of H is an even permutation or exactly half of the members are even. (This exercise is referred to in Chapter 25.)

Problem 9E Prove that the mapping from G ? H to G given by (g, h) S g is a homomorphism. What is the kernel? This mapping is called the projection of G ? H onto G.

Problem 10E Let G be a subgroup of some dihedral group. For each x in G, define Prove that f is a homomorphism from G to the multiplicative group {+1, –1}. What is the kernel? Why does this prove Exercise 25 of Chapter 3?

Problem 11E Prove that is isomorphic to Za ? Zb.

Problem 12E Suppose that k is a divisor of n. Prove that .

Problem 14E Explain why the correspondence x ? 3x from Z12 to Z10 is not a homomorphism.

Problem 15E Suppose that f is a homomorphism from Z30 to Z30 and Ker ? 5 {0, 10, 20}. If ?(23) = 9, determine all elements that map to 9.

Problem 16E Prove that there is no homomorphism from Z8 ? Z2 onto Z4 ? Z4.

Problem 13E Prove that (A ? B)/(A ? {e}) ? B.

Problem 17E Prove that there is no homomorphism from Z16 ? Z2 onto Z4 ? Z4.

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.

Problem 19E Suppose that there is a homomorphism f from Z17 to some group and that ? is not one-to-one. Determine ?.

Problem 20E How many homomorphisms are there from Z20 onto Z8? How many are there to Z8?

Problem 21E If ? is a homomorphism from Z30 onto a group of order 5, determine the kernel of ?.

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.

Suppose that \(\phi\) is a homomorphism from \(Z_{36}\) to a group of order 24. a. Determine the possible homomorphic images. b. For each image in part a, determine the corresponding kernel of \(\phi\).

Problem 24E Suppose that ?: Z50 S Z15 is a group homomorphism with ? (7) = 6. a. Determine ? (x). b. Determine the image of ?. c. Determine the kernel of?. d. Determine ?21(3). That is, determine the set of all elements that map to 3.

Problem 25E How many homomorphisms are there from Z20 onto Z10? How many are there to Z10?

Problem 26E Determine all homomorphisms from Z4 to Z2 ? Z2.

Problem 27E Determine all homomorphisms from Zn to itself.

Problem 29E Suppose that there is a homomorphism from a finite group G onto Z10. Prove that G has normal subgroups of indexes 2 and 5.

Problem 28E Suppose that f is a homomorphism from S4 onto Z2. Determine Ker ?. Determine all homomorphisms from S4 to Z2.

Problem 31E Suppose that ? is a homomorphism from U(30) to U(30) and that Ker ? = {1, 11}. If ?(7) = 7, find all elements of U(30) that map to 7.

Problem 32E Find a homomorphism ? from U(30) to U(30) with kernel {1, 11} and ?(7) = 7.

Problem 30E Suppose that f is a homomorphism from a group G onto Z6 ? Z2 and that the kernel of ? has order 5. Explain why G must have normal subgroups of orders 5, 10, 15, 20, 30, and 60.

Problem 33E Suppose that ? is a homomorphism from U(40) to U(40) and that Ker ? = {1, 9, 17, 33}. If ?(11) = 11, find all elements of U(40) that map to 11.

Problem 34E Find a homomorphism f from U(40) to U(40) with kernel {1, 9, 17, 33} and ?(11) = 11.

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).

Problem 36E Suppose that there is a homomorphism f from Z?Z to a group G such that ?((3, 2)) = a and ?((2, 1)) = b. Determine ?((4, 4)) in terms of a and b. Assume that the operation of G is addition.

Problem 38E Let a be a homomorphism from G1 to H1 and b be a homomorphism from G2 to H2. Determine the kernel of the homomorphism g from G1 ? G2 to H1 ? H2 defined by g(g1, g2) = (a(g1), b(g2)).

Problem 37E Let H = {z ? C* | |z| = 1}. Prove that C*/H is isomorphic to R+, the group of positive real numbers under multiplication.

Problem 40E For each pair of positive integers m and n, we can define a homomorphism from Z to Zm ? Znby x ? (x mod m, x mod n). What is the kernel when (m, n) = (3, 4)? What is the kernel when (m, n) = (6, 4)? Generalize.

Problem 39E Prove that the mapping x ? x6 from C* to C* is a homomorphism. What is the kernel?

Problem 42E (Third Isomorphism Theorem) If M and N are normal subgroups of G and N ? M, prove that (G/N)/(M/N) L G/M.

Problem 41E (Second Isomorphism Theorem) If K is a subgroup of G and N is a normal subgroup of G, prove that K/(K ? N) is isomorphic to KN/N.

Problem 43E Let ?(d) denote the Euler phi function of d (see page 85). Show that the number of homomorphisms from Zn to Zk is ??(d), where the sum runs over all common divisors d of n and k. [It follows from number theory that this sum is actually gcd(n, k).]

Problem 45E Determine all homomorphic images of D4 (up to isomorphism).

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)?

Problem 46E 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.

Problem 47E Suppose that G is a finite group and that Z10 is a homomorphic image of G. What can we say about |G|? Generalize.

Problem 49E Suppose that for each prime p, Zp is the homomorphic image of a group G. What can we say about |G|? Give an example of such a group.

Problem 50E (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. Reference: Theorem 10.1 Properties of Elements Under Homomorphisms

Suppose that \(Z_{10}\) and \(Z_{15}\) are both homomorphic images of a finite group G. What can be said about |G|? Generalize.

Show that a homomorphism defined on a cyclic group is completely determined by its action on a generator of the group.

Problem 53E Use the First Isomorphism Theorem to prove Theorem 9.4. Reference:

Let \(\alpha\) and \(\beta\) be group homomorphisms from G to \(\bar{G}\) and let \(H= \{g \in G \mid \alpha(g)=\beta(g)\}\). Prove or disprove that H is a subgroup of G

Problem 56E Prove that the mapping from R under addition to GL(2, R) that takes x to is a group homomorphism. What is the kernel of the homomorphism?

Problem 55E 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) ? f(3) is a homomorphism. Give a geometric description of the kernel of this homomorphism. Generalize.

Problem 51E 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 24.) Reference: Theorem 10.2 Properties of Subgroups Under Homomorphisms

Problem 57E Suppose there is a homomorphism f from G onto Z2 ? Z2. Prove that G is the union of three proper normal subgroups.

Problem 58E If H and K are normal subgroups of G and H ? K = {e}, prove that G is isomorphic to a subgroup of G/H ? G/K.

Problem 59E Suppose that H and K are distinct subgroups of G of index 2. Prove that H ? K is a normal subgroup of G of index 4 and that G/(H ? K) is not cyclic.

Problem 61E Prove that every group of order 77 is cyclic.

Determine all homomorphisms from Z onto \(S_3\). Determine all homomorphisms from Z to \(S_3\).

Problem 63E Let G be an Abelian group. Determine all homomorphisms from S3 to G.

Problem 60E Suppose that the number of homomorphisms from G to H is n. How many homomorphisms are there from G to H ? H ? … ? H (s terms)? When H is Abelian, how many homomorphisms are there from G ? G ? …? G (s terms) to H?

Problem 65E Prove that the mapping from C* to C* given by ? (z) = z2 is a homomorphism and that C*/ {1, –1} is isomorphic to C*.

If \(\phi\) is an isomorphism from a group G under addition to a group \(\bar{G}\) under addition, prove that for any integer n, the mapping from G to \(\bar{G}\) defined by \(\gamma(x)=n \phi(x)\) is a homomorphism from G to \(\bar{G}\).

Let p be a prime. Determine the number of homomorphisms from \(Z_p \oplus Z_p\) into \(Z_p\).