Use the results presented in this chapter to prove that

Problem 89E Assuming that a message has been sent via the RSA scheme with p = 37, q = 73, and e = 5, decode the received message “34.”

Problem 1E Prove that the external direct product of any finite number of groups is a group. (This exercise is referred to in this chapter.)

Problem 1SE A subgroup N of a group G is called a characteristic subgroup if ?(N) 5 N for all automorphisms ? of G. (The term characteristic was first applied by G. Frobenius in 1895.) Prove that every subgroup of a cyclic group is characteristic.

Problem 1CE This software lists the elements of Us(st), where s and t are relatively prime. Run the program for (s, t) = (5, 16), (16, 5) , (8, 25), (5, 9), (9, 5), (9, 10), (10, 9), and (10, 25).

Problem 2E Show that Z2 ? Z2 ? Z2 has seven subgroups of order 2.

Problem 2CE Exercise 2. This software computes the elements of the subgroup U(n)k = {xk | x U(n)} of U(n) and its order. Run the program for (n,k) = (27,3), (27,5), (27,7), and (27,11). Do you see a relationship connecting|U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (25,3), (25,5), (25,7), and (25,11). Do you see a relationship connecting |U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (32,2), (32,4), and (32,8). Is your conjecture valid for U(32,16)? If not, restrict your conjecture. Run the program for (n,k) = (77,2), (77,3), (77,5), (77,6), (77,10), and (77,15)? Do you see a relationship among U(77,6) and U(77,2), and U(77,3)? What about U(77,10), U(77,2), and U(77,5)? What about U(77,15), U(77,3), and U(77,5)? Make a conjecture. Use the theory developed in this chapter about expressing U(n) as external direct products of cyclic groups of the form Zn to analyze these groups to verify your conjectures. Exercise 3. This software implements the algorithm given in Chapter 8 to express U(n) as an external direct product of groups of the form Zk. Assume that n is given in prime-power factorization form. Run your program for 3 . 5 . 7, 16 . 9 . 5, 8 . 3 . 25, 9 . 5 . 11, and 2 . 27 . 125. [ NOTE: Please enter the prime-power factorization form with a `period(".")' in between the integers and without any space. Also, this program has been written to accept n as any integer, i.e., instead of entering n in the factored form as 3 . 5 . 7 you could enter 105 . ] Exercise 5. This program implements the RSA public key cryptography scheme. The user enters two primes p and q, an r that is relatively prime to m = lcm(p -1,q -1), and the message M to be sent. Then the program computes the s which is the inverse of r mod m, and the value of Mr mod pq. Then the user can input those numbers and have the computer raise the numbers to the s power to obtain the original input.

Problem 3CE Exercise 2. This software computes the elements of the subgroup U(n)k = {xk | x U(n)} of U(n) and its order. Run the program for (n,k) = (27,3), (27,5), (27,7), and (27,11). Do you see a relationship connecting|U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (25,3), (25,5), (25,7), and (25,11). Do you see a relationship connecting |U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (32,2), (32,4), and (32,8). Is your conjecture valid for U(32,16)? If not, restrict your conjecture. Run the program for (n,k) = (77,2), (77,3), (77,5), (77,6), (77,10), and (77,15)? Do you see a relationship among U(77,6) and U(77,2), and U(77,3)? What about U(77,10), U(77,2), and U(77,5)? What about U(77,15), U(77,3), and U(77,5)? Make a conjecture. Use the theory developed in this chapter about expressing U(n) as external direct products of cyclic groups of the form Zn to analyze these groups to verify your conjectures. Exercise 3. This software implements the algorithm given in Chapter 8 to express U(n) as an external direct product of groups of the form Zk. Assume that n is given in prime-power factorization form. Run your program for 3 . 5 . 7, 16 . 9 . 5, 8 . 3 . 25, 9 . 5 . 11, and 2 . 27 . 125. [ NOTE: Please enter the prime-power factorization form with a `period(".")' in between the integers and without any space. Also, this program has been written to accept n as any integer, i.e., instead of entering n in the factored form as 3 . 5 . 7 you could enter 105 . ] Exercise 5. This program implements the RSA public key cryptography scheme. The user enters two primes p and q, an r that is relatively prime to m = lcm(p -1,q -1), and the message M to be sent. Then the program computes the s which is the inverse of r mod m, and the value of Mr mod pq. Then the user can input those numbers and have the computer raise the numbers to the s power to obtain the original input.

Problem 2SE Prove that the center of a group is characteristic.

Let G be a group with identity \(e_G\) and let H be a group with identity \(e_H\). Prove that G is isomorphic to \(G \oplus \{e_H\}\) and that H is isomorphic to \(\{e_G\} \oplus H\).

Problem 3SE The commutator subgroup G' of a group G is the subgroup generated by the set {x–1y–1xy | x, y ? G}. (That is, every element of G' has the form a1 i1a2 i2 … ak ik, where each aj has the form x–1y–1xy, each ij = ±1, and k is any positive integer.) Prove that G' is a characteristic subgroup of G. (This subgroup was first introduced by G. A. Miller in 1898.)

Problem 4CE This software allows you to input positive integers n_1, n_2>, n_3, ... , n_k, where k <= 5, and compute the number of elements in Zn_1+Zn_2+...+Zn_k of any specified order m. Use this software to verify the values obtained in Examples 3 and 4. Run the software for n_1=6, n_2=10, n_3=12, and m=6. Top of Form Please input n_1, n_2, ... , n_k here: Bottom of Form (Example: if n_1=6, n_2=10, n_3=12, you may input 6, 10, 12) Please input order m here:

Problem 4SE Prove that the property of being a characteristic subgroup is transitive. That is, if N is a characteristic subgroup of K and K is a characteristic subgroup of G, then N is a characteristic subgroup of G.

Problem 4E Show that G ? H is Abelian if and only if G and H are Abelian. State the general case.

This program implements the RSA public key cryptography scheme. The user enters two primes p and q, an r that is relatively prime to m = lcm(p -1,q -1), and the message M to be sent. Then the program computes the s which is the inverse of r mod m, and the value of \(M^r\) mod pq. Then the user can input those numbers and have the computer raise the numbers to the s power to obtain the original input. Go to https://www.d.umn.edu/~jgallian/compsciProject2018/html/chap8/ch8ex5.html

Problem 5E Prove or disprove that Z ? Z is a cyclic group.

Problem 5SE Let and let H be the subgroup of SL(3, Z3) consisting of (See Exercise 48 in Chapter 2 for the definition of multiplication.) Determine the number of elements of each order in G and H. Are G and H isomorphic? (This exercise shows that two groups with the same number of elements of each order need not be isomorphic.) Reference:

Problem 6CE This software determines the order of , where p is a prime less than 101. Run the software for p = 3, 5, and 7. Is the result always divisible by p? Is the result always divisible by p-1? Is the result always divisible by p+1? Make a conjecture about the order of for all primes p.

Problem 6E Prove, by comparing orders of elements, that Z8 ?Z2 is not isomorphic to Z4 ? Z4.

Problem 6SE Let H and K be subgroups of a group G and let HK = {hk | h?H, k?K} and KH = {kh | k ? K, h ? H}. Prove that HK is a group if and only if HK = KH.

Problem 7CE This software determines the order of , where p is a prime less than 101. Run the software for p = 3, 5, and 7. What is the highest power of p that divides the order? What is the highest power of p-1 that divides the order? What is the highest power of p+1 that divides the order? Make a conjecture about the order of for all primes p

Problem 7E Prove that G1 ? G2 is isomorphic to G2 ? G1. State the general case.

Let G be a finite Abelian group in which every nonidentity element has order 2. If |G| > 2, prove that the product of all the elements in G is the identity.

Problem 8CE This software determines the order of , where p is a prime less than 101. Run the software for p = 3, 5, and 7. What is the highest power of p that divides the order? What is the highest power of p-1 that divides the order? Make a conjecture about the order of for all primes p.

Problem 8E Is Z3 ?Z9 isomorphic to Z27? Why?

Prove that \(S_4\) is not isomorphic to \(D_{4} \oplus Z_{3}\).

Problem 9CE This software determines the order of , where p is a prime less than 101. Run the software for p = 3, 5, and 7. What is the highest power of p that divides the order? What is the highest power of p-1 that divides the order? Make a conjecture about the order of for all primes p.

Is \(Z_3 \oplus Z_5\) isomorphic to \(Z_{15}\)? Why?

Problem 9SE Let G be a group. For any element g of G, define gZ(G) = {gh | h ? Z(G)}. If a is an element of G of order 4, prove that H = Z(G) x aZ(G) x a2Z(G) x a3Z(G) is a subgroup of G. Generalize to the case that |a| = k.

Problem 10CE This software determines the order of , where p is a prime less than 71. Run the software for p = 3, 5, 7 and 11. What is the highest power of p that divides the order? What is the highest power of p-1 that divides the order? Make a conjecture about the order of for all primes p.

Problem 10E How many elements of order 9 does Z3 ? Z9 have? (Do not do this exercise by brute force.)

The exponent of a group is the smallest positive integer n such that \(x^n = e\) for all x in the group. Prove that every finite group has an exponent that divides the order of the group.

Problem 11E How many elements of order 4 does Z4 ? Z4 have? (Do not do this by examining each element.) Explain why Z4 ? Z4 has the same number of elements of order 4 as does Z8000000 ? Z400000. Generalize to the case Zm ? Zn.

Problem 11SE Determine all U-groups of exponent 2.

Problem 12E Give examples of four groups of order 12, no two of which are isomorphic. Give reasons why no two are isomorphic.

Problem 12SE Suppose that H and K are subgroups of a group and that |H| and |K| are relatively prime. Show that H?K = {e}.

Problem 13E For each integer n > 1, give examples of two nonisomorphic groups of order n2.

Let \(\mathbf{R}^{+}\) denote the multiplicative group of positive real numbers and let \(T=\left\{a+b i \in \mathbf{C}^* \mid a^2+b^2=1\right\}\) be the multiplicative group of complex numbers on the unit circle. Show that every element of \(\mathbf{C}^*\) can be uniquely expressed in the form rz, where \(r \in \mathbf{R}^{+}\) and \(z \in T\).

Problem 14E The dihedral group Dn of order 2n (n ? 3) has a subgroup of n rotations and a subgroup of order 2. Explain why Dn cannot be isomorphic to the external direct product of two such groups.

Problem 14SE Prove that Q* under multiplication is not isomorphic to R* under multiplication.

Problem 15E Prove that the group of complex numbers under addition is isomorphic to R ? R.

Problem 15SE Prove that Q under addition is not isomorphic to R under addition.

Suppose that \(G_1 \approx G_2\) and \(H_1 \approx H_2\). Prove that \(G_1 \oplus H_1 \approx G_2 \oplus\) \(\mathrm{H}_2\). State the general case.

Prove that R under addition is not isomorphic to R* under multiplication.

Problem 17E If G ? H is cyclic, prove that G and H are cyclic. State the general case.

Show that \(Q^+\) (the set of positive rational numbers) under multiplication is not isomorphic to Q under addition.

In \(Z_{40} \oplus Z_{30}\), find two subgroups of order 12.

Problem 18SE Suppose that G = {e, x, x2, y, yx, yx2} is a non-Abelian group with |x| = 3 and |y| = 2. Show that xy = yx2.

If r is a divisor of m and s is a divisor of n, find a subgroup of \(Z_m \oplus\) \(Z_n\) that is isomorphic to \(Z_r \oplus Z_s\).

Problem 19SE Let p be an odd prime. Show that 1 is the only solution of xp–2 = 1 in U(p).

Problem 20E Find a subgroup of Z12 ? Z18 that is isomorphic to Z9 ? Z4.

Let G be an Abelian group under addition. Let n be a fixed positive integer and let \(H=\{(g, n g) \mid g \in G\}\). Show that H is a subgroup of \(G \oplus G\). When G is the set of real numbers under addition, describe H geometrically.

Problem 21E Let G and H be finite groups and (g, h) ?G ? H. State a necessary and sufficient condition for .

Find a subgroup of \(Z_{12} \oplus Z_{20}\) that is isomorphic to \(Z_{4} \oplus Z_{5}\).

Problem 22E Determine the number of elements of order 15 and the number of cyclic subgroups of order 15 in Z30 ? Z20.

Suppose that \(G=G_{1} \oplus G_{2} \oplus \cdots \oplus G_{n}\). Prove that \(Z(G)=Z\left(G_{1}\right) \oplus Z\left(G_{2}\right) \oplus \cdots \oplus Z\left(G_{n}\right)\).

Problem 23E What is the order of any nonidentity element of Z3 ? Z3 ? Z3? Generalize.

Problem 23SE Exhibit four nonisomorphic groups of order 18.

Problem 24E Let m >2 be an even integer and let n > 2 be an odd integer. Find a formula for the number of elements of order 2 in Dm ? Dn.

What is the order of the largest cyclic subgroup in Aut\((Z_{720})\)? (Hint: It is not necessary to consider automorphisms of \(Z_{720}\).)

Problem 25E Let M be the group of all real 2 × 2 matrices under addition. Let N = R ? R ? R ? R under component wise addition. Prove that M and N are isomorphic. What is the corresponding theorem for the group of m × n matrices under addition?

Problem 25SE Let G be the group of all permutations of the positive integers. Let H be the subset of elements of G that can be expressed as a product of a finite number of cycles. Prove that H is a subgroup of G.

The group \(S_3 \oplus Z_2\) is isomorphic to one of the following groups: \(Z_{12}, Z_6 \oplus Z_2, A_4, D_6\). Determine which one by elimination.

Let G be a group and let \(g \in G\). Show that \(Z(G)\langle g\rangle\) is a subgroup of G.

Problem 27E Let G be a group, and let H = {(g, g) | g?G}. Show that H is a subgroup of G ? G. (This subgroup is called the diagonal of G ? G.) When G is the set of real numbers under addition, describe G ? G and H geometrically.

Show that \(D_{11} \oplus Z_{3} \not\approx D_{3} \oplus Z_{11}\).

Problem 28E Find a subgroup of Z4 ? Z2 that is not of the form H ? K, where H is a subgroup of Z4 and K is a subgroup of Z2.

Problem 28SE Show that . (This exercise is referred to in Chapter 24.)

Find all subgroups of order 3 in \(Z_{9} \oplus Z_{3}\).

Problem 30E Find all subgroups of order 4 in Z4 ? Z4.

Problem 30SE Exhibit four nonisomorphic groups of order 66. (This exercise is referred to in Chapter 24.)

Problem 31E What is the largest order of any element in Z30 ? Z20?

Problem 29SE Show that . (This exercise is referred to in Chapter 24.)

Problem 31SE Prove that |Inn(G)| = 1 if and only if G is Abelian.

Problem 32E What is the order of the largest cyclic subgroup of Z6 ? Z10 ? Z15? What is the order of the largest cyclic subgroup of Zn1? Zn2? …? Znk?

List four elements of \(Z_{20} \oplus Z_5 \oplus Z_{60}\) that form a noncyclic subgroup.

Find a subgroup of order 6 in U(450).

Prove that \(x^{100} = 1\) for all x in U(1000).

Find three cyclic subgroups of maximum possible order in \(Z_6 \oplus Z_{10} \oplus Z_{15}\) of the form \(\langle a\rangle \oplus\langle b\rangle \oplus\langle c\rangle\), where \(a \in Z_{6}, b \in Z_{10}\), and \(c \in Z_{15}\).

Problem 35E Find a subgroup of Z800 ? Z200 that is isomorphic to Z2 ? Z4.

Problem 34E How many elements of order 2 are in Z2000000 ? Z4000000? Generalize.

Problem 35SE In S10, let ? = (13)(17)(265)(289). Find an element in S10 that commutes with ? but is not a power of ?.

Problem 36E Find a subgroup of Z12 ? Z4 ? Z15 that has order 9.

Problem 36SE Prove or disprove that

Problem 37E Prove that R* ? R* is not isomorphic to C*. (Compare this with Exercise 15.) Reference: Prove that the group of complex numbers under addition is isomorphic to R ? R.

Prove or disprove that \(D_{12} \approx Z_3 \oplus D_4\).

Let \(H=\left\{\left[\begin{array}{lll}1 & a & b \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right] \mid a, b \in Z_{3}\right\}\) (See Exercise 48 in Chapter 2 for the definition of multiplication.) Show that H is an Abelian group of order 9. Is H isomorphic to \(Z_{9}\) or to \(Z_{3} \oplus Z_{3}\)?

Problem 38SE Describe a three-dimensional solid whose symmetry group is isomorphic to D5.

Problem 39E Let G = {3m6n | m, n ? Z} under multiplication. Prove that G is isomorphic to Z ? Z. Does your proof remain valid if G = {3m9n | m, n ? Z}?

Problem 39SE Let . Find the order of (2, 3, (123)(15)). Find the inverse of (2, 3, (123)(15)).

Problem 40E Let (a1, a2, . . . , an) ? G1 ? G2 ? … ? Gn. Give a necessary and sufficient condition for |(a1, a2, . . . , an)| = ?

Problem 40SE Let and let H = {g ? G | |g| = ? or |g| = 1}. Prove or disprove that H is a subgroup of G.

Problem 41SE Find a subgroup H of is isomorphic to

Prove that \(D_3 \oplus D_4 \neq D_{12} \oplus Z_2\).

Problem 42E Determine the number of cyclic subgroups of order 15 in Z90 ? Z36. Provide a generator for each of the subgroups of order 15.

Problem 43E List the elements in the groups U5(35) and U7(35).

Problem 43SE Find an element of order 10 in A9.

Problem 42SE Find three subgroups H1, H2, and H3 of such that ( )/Hi is isomorphic to Zp2 for i = 1, 2, 3.

Problem 44E Prove or disprove that U(40) ? Z6 is isomorphic to U(72) ? Z4.

In the left regular representation for \(D_4\), write \(T_{R_{90}}\) and \(T_H\) in matrix form and in cycle form.

Problem 45E Prove or disprove that C* has a subgroup isomorphic to Z2 ? Z2.

How many elements of order 6 are in \(S_7\)?

Problem 46E Let G be a group isomorphic to Zn1 ? Zn2 ? . . . ? Znk. Let x be the product of all elements in G. Describe all possibilities for x.

Problem 46SE Prove that is not isomorphic to a subgroup of S6.

Problem 47E If a group has exactly 24 elements of order 6, how many cyclic subgroups of order 6 does it have?

Find a permutation \(\beta\) such that \(\beta^2 = (13579)(268)\).

Problem 48E For any Abelian group G and any positive integer n, let Gn = {gn | g ? G} (see Exercise 17, Supplementary Exercises for Chapters 1– 4). If H and K are Abelian, show that (H ? K)n 5 Hn ? Kn. Reference: Let G be an Abelian group and let n be a fixed positive integer. Let Gn = {gn | g ? G}. Prove that Gn is a subgroup of G. Give an example showing that Gn need not be a subgroup of G when G is non-Abelian. (This exercise is referred to in Chapter 11.)

Problem 49E Express Aut(U(25)) in the form Zm ? Zn.

In \(\mathbf R \oplus \mathbf R\) under componentwise addition, let \(H = \{(x, 3x) | x \in \mathbf R\}\). (Note that H is the subgroup of all points on the line y = 3x.) Show that (2, 5) + H is a straight line passing through the point (2, 5) and parallel to the line y = 3x.

Problem 49SE In , suppose that H is the subgroup of all points lying on a line through the origin. Show that any left coset of H is a line parallel to H.

Problem 50E Determine Aut(Z2 ? Z2).

Suppose that \(n_1, n_2, \ldots , n_k\) are positive even integers. How many elements of order 2 does \(Z_{n_1} \oplus Z_{n_2} \oplus \ldots \oplus Z_{n_k}\) have? How many are there if we drop the requirement that \(n_1, n_2, \ldots , n_k\) must be even?

Let G be a group of permutations on the set \(\{1,2, \ldots, n\}\). Recall that \(\operatorname{stab}_G(1)=\{\alpha \in G \mid \alpha(1)=1\}\). If \(\gamma\) sends 1 to k, prove that \(\gamma \operatorname{stab}_G(1)=\{\beta \in G \mid \beta(1)=k\}\)

Let H be a subgroup of G and let \(a, b \in G\). Show that aH = bH if and only if \(Ha^{-1} = Hb^{-1}\).

Problem 52E Is Z10 ? Z12 ? Z6 ? Z60 ? Z6 ? Z2?

Problem 52SE Suppose that G is a finite Abelian group that does not contain a subgroup isomorphic to for any prime p. Prove that G is cyclic.

Problem 53SE Let p be a prime. Determine the number of elements of order p in

Problem 53E Is Z10 ? Z12 ? Z6 ? Z15 ? Z4 ? Z12?

Problem 54SE Show that .

Problem 55E How many isomorphisms are there from Z12 to Z4 ? Z3?

Problem 54E Find an isomorphism from Z12 to Z4 ? Z3.

Let p be a prime. Determine the number of subgroups of \(Z_{p^2} \oplus Z_{p^2}\) that are isomorphic to \(Z_{p^2}\).

Problem 56E Suppose that ? is an isomorphism from Z3 ? Z= to Z15 and ? (2, 3) 5 2. Find the element in Z3 ? Z5 that maps to 1.

Problem 56SE Find a group of order that contains a subgroup isomorphic to A8.

Problem 57SE Let p and q be distinct odd primes. Let n = lcm(p – 1, q – 1). Prove that xn = 1 for all x?U( pq).

Problem 58E Prove that Z5 ? Z5 has exactly six subgroups of order 5.

Problem 57E If f is an isomorphism from Z4 ? Z3 to Z12, what is ? (2, 0)? What are the possibilities for ? (1, 0)? Give reasons for your answer.

Let (a, b) belong to \(Z_{m} \oplus Z_{n}\). Prove that |(a, b)| divides lcm(m, n).

Problem 59SE Prove that the permutations (12) and (123 . . . n) generate Sn. (That is, every member of Sncan be expressed as some combination of these elements.)

Problem 60E Let G = {ax2 1 bx 1 c | a, b, c ? Z3}. Add elements of G as you would polynomials with integer coefficients, except use modulo 3 addition. Prove that G is isomorphic to Z3 ? Z3 ? Z3. Generalize.

Problem 58SE Give a simple characterization of all positive integers n for which for every subgroup H of Zn.

Problem 60SE Suppose that n is even and s is an (n – 1)-cycle in Sn. Show that s does not commute with any element of order 2.

Determine all cyclic groups that have exactly two generators.

Suppose that n is odd and \(\sigma\) is an n-cycle in \(S_n\). Prove that \(\sigma\) does not commute with any element of order 2.

Problem 62E Explain a way that a string of length n of the four nitrogen bases A, T, G, and C could be modeled with the external direct product of n copies of Z2 ? Z2.

Problem 63SE Let m be a positive integer. For any n-cycle ?, show that ?m is the product of gcd(m, n) disjoint cycles, each of length n/gcd(m, n).

Give an example of an infinite non-Abelian group that has exactly six elements of finite order.

Problem 62SE Let H = {? ? Sn | a maps the set {1, 2} to itself}. Prove that C ((12)) = H.

Let p be a prime. Prove that \(Z_{p} \oplus Z_{p}\) has exactly p + 1 subgroups of order p.

Problem 65E Give an example to show that there exists a group with elements a and b such that |

Express U(165) as an external direct product of U-groups in four different ways.

Problem 66E Express U(165) as an external direct product of cyclic groups of the form Zn.

Problem 68E Without doing any calculations in Aut(Z20), determine how many elements of Aut(Z20) have order 4. How many have order 2?

Problem 70E Without doing any calculations in U(27), decide how many subgroups U(27) has.

Problem 69E Without doing any calculations in Aut(Z720), determine how many elements of Aut(Z720) have order 6.

Problem 71E What is the largest order of any element in U(900)?

Problem 72E Let p and q be odd primes and let m and n be positive integers. Explain why U( pm) ? U(qn) is not cyclic.

Problem 73E Use the results presented in this chapter to prove that U(55) is isomorphic to U(75).

Problem 74E Use the results presented in this chapter to prove that U(144) is isomorphic to U(140).

Problem 75E For every n > 2, prove that U(n)2 = {x2 | x ? U(n)} is a proper subgroup of U(n).

Problem 76E Show that U(55)3 = {x3 | x ? U(55)} is U(55).

Problem 77E Find an integer n such that U(n) contains a subgroup isomorphic to Z5 ? Z5.

Problem 78E Find a subgroup of order 6 in U(700).

Problem 79E Show that there is a U-group containing a subgroup isomorphic to Z3 ? Z3.

Problem 80E Find an integer n such that U(n) is isomorphic to Z2 ? Z4 ? Z9.

Problem 82E If k divides m and m divides n, how are Um(n) and Uk(n) related?

What is the smallest positive integer k such that \(x^k =e\) for all x in \(U(7 \cdot 17)\)? Generalize to U(pq) where p and q are distinct primes.

Problem 83E Let p1, p2,…, pk be distinct odd primes and n1, n2,…, nk be positive integers. Determine the number of elements of order 2 in where n is at least 3?

Problem 84E Show that no U-group has order 14.

Show that there is a U-group containing a subgroup isomorphic to \(Z_{14}\).

Problem 88E Using the RSA scheme with p = 37, q = 73, and e = 5, what number would be sent for the message “RM”?

Problem 86E Show that no U-group is isomorphic to Z4 ? Z4.

Show that there is a U-group containing a subgroup isomorphic to \(Z_4 \oplus Z_4\)