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All Textbook Solutions for Contemporary Abstract Algebra
For n=5 , 8, 12, 20, and 25, find all positive integers less than n and relatively prime to n.Determine a. gcd(2,10) lcm(2,10) b. gcd(20,8) lcm(20,8) d. gcd(21,50) lcm(21,50) e. gcd (p2q2,pq3) lcm (p2q2,pq3) where p and q are distinct primesDetermine 51 mod 13, 342 mod 85, 62 mod 15, 10 mod 15, (8273) mod 7, (51+68) mod 7, (3524) mod 11, and (47+68) mod 11.Find integers s and t such that 1=7s+11t ? t. Show that s and t are not unique.Show that if a and b are positive integers, then ab=lcm(a,b)gcd(a,b) .Suppose a and b are integers that divide the integer c. If a and b are relatively prime, show that ab divides c. Show, by example, that if a and b are not relatively prime, then ab need not divide c.If a and b are integers and n is a positive integer, prove that a mod n=b mod n if and only if n divides ab .Let d=gcd(a,b) . If a=da and b=db , show that gcd(a,b)=1 .Let n be a fixed positive integer greater than 1. If a mod n=a and b mod n=b , prove that (a+b)modn=(a+b) mod n and (ab) mod n=(ab) mod n. (This exercise is referred to in Chapters 6,8, 10, and 15.)Let a and b be positive integers and let d=gcd(a,b) and m=lcm(a,b) . If t divides both a and b, prove that t divides d. If s is amultiple of both a and b, prove that s is a multiple of m.Let n and a be positive integers and let d=gcd(a,n) . Show that the equation ax mod n=1 has a solution if and only if d=1 . (This exercise is referred to in Chapter 2.)Show that 5n+3and7n+4 are relatively prime for all n.Suppose that m and n are relatively prime and r is any integer. Showthat there are integers x and y such that mx+ny=r .Let p, q, and r be primes other than 3. Show that 3 divides p2+q2+r2 .Prove that every prime greater than 3 can be written in the form 6n+1or6n+5 .Determine 71000 mod 6 and 61001 mod 7.Let a, b, s, and t be integers. If a mod st=bmodst, show that amod s=bmods and a mod t=bmodt. What condition on s and tis needed to make the converse true? (This exercise is referred to in Chapter 8.)Determine 8402 mod 5.Show that gcd(a,bc)=1 if and only if gcd(a,b)=1 and gcd(a,c)=1 . (This exercise is referred to in Chapter 8.)Let p1,p2,...,pn be primes. Show that p1p2pn+1 is divisible by none of these primes.Prove that there are infinitely many primes. (Hint: Use Exercise 20.)22E23EFor any complex numbers z1andz2 prove that z1z2=z1z2 .Give an “if and only if” statement that describes when the logic gatex NAND y modeled by 1+xy is 1. Give an “if and only if” statement that describes when the logic gate x XNOR y modeled by 1+x+y is 1.For inputs of 0 and 1 and mod 2 arithmetic describe the output ofthe formula z+xy+xz in the form “If x … , else …”.For every positive integer n, prove that a set with exactly n elements has exactly 2n subsets (counting the empty set and the entire set).Prove that 2n32n1 is always divisible by 17.Prove that there is some positive integer n such that n, n+1,n+2, , n+200 are all composite.(Generalized Euclid’s Lemma) If p is a prime and p divides a1a2an , prove that p divides ai for some i.31EWhat is the largest bet that cannot be made with chips worth $7.00 and $9.00? Verify that your answer is correct with both forms of induction.Prove that the First Principle of Mathematical Induction is a consequence of the Well Ordering Principle.The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . . In general,the Fibonacci numbers are defined by f1=1,f2=1 , and for n3,fn=fn1+fn2 . Prove that the nth Fibonacci number fn satisfies fn2n .Prove by induction on n that for all positive integers n, n3+(n+1)3+(n+2)3 is a multiple of 9.Suppose that there is a statement involving a positive integer parameter n and you have an argument that shows that whenever the statement is true for a particular n it is also true for n+2 . What remains to be done to prove the statement is true for every positive integer? Describe a situation in which this strategy would beapplicable.In the cut “As” from Songs in the Key of Life, Stevie Wonder mentions the equation 888=4 . Find all integers n for which this statement is true, modulo n.Prove that for every integer n, n3 mod 6=n mod 6.If it is 2:00 A.M. now, what time will it be 3736 hours from now?Determine the check digit for a money order with identification number 7234541780.Suppose that in one of the noncheck positions of a money order number, the digit 0 is substituted for the digit 9 or vice versa. Prove that this error will not be detected by the check digit. Prove that all other errors involving a single position are detected.Suppose that a money order identification number and check digit of 21720421168 is erroneously copied as 27750421168. Will the check digit detect the error?A transposition error involving distinct adjacent digits is one of theform ...ab......ba... with ab . Prove that the money order check-digit scheme will not detect such errors unless the check digititself is transposed.Determine the check digit for the Avis rental car with identification number 540047. (See Example 5.)Show that a substitution of a digit ai for the digit ai(aiai) in a noncheck position of a UPS number is detected if and only if aiai7 .Determine which transposition errors involving adjacent digits are detected by the UPS check digit.Use the UPC scheme to determine the check digit for the number 07312400508.Explain why the check digit for a money order for the number N is the repeated decimal digit in the real number N9 .The 10-digit International Standard Book Number (ISBN-10) a1a2a3a4a5a6a7a8a9a10 has the property (a1,a2,...,a10)(10,9,8,7,6,5,4,3,2,1) mod 11=0 . The digit a10 is the check digit. When a10is required to be 10 to make the dot product 0, the character X is used as the check digit. Verify the check digit for the ISBN-10 assigned tothis book.Suppose that an ISBN-10 has a smudged entry where the question mark appears in the number 0716?28419 . Determine the missing digit.Suppose three consecutive digits abc of an ISBN-10 are scrambled as bca. Which such errors will go undetected?52ESuppose the weighting vector for ISBN-10s were changed to (1, 2,3, 4, 5, 6, 7, 8, 9, 10). Explain how this would affect the check digit.Use the two-check-digit error-correction method described in thischapter to append two check digits to the number 73445860.Suppose that an eight-digit number has two check digits appended using the error-correction method described in this chapter and it is incorrectly transcribed as 4302511568. If exactly one digit is incorrect,determine the correct number.The state of Utah appends a ninth digit a9 to an eight-digit driver’slicense number a1a2...a8 so that (9a1+8a2+7a3+6a4+5a5+4a6+3a7+2a8+a9)mod10=0 . If you know that the license number 149105267 has exactly one digit incorrect, explain why the error cannot be in position 2, 4, 6, or 8.Complete the proof of Theorem 0.8.Let S be the set of real numbers. If a,bS , define a~b if ab isan integer. Show that ~is an equivalence relation on S. Describe the equivalence classes of S.Let S be the set of integers. If a,bS , define aRb if ab0 . Is R an equivalence relation on S?Let S be the set of integers. If a,bS , define aRb if a+b is even. Prove that R is an equivalence relation and determine the equivalence classes of S.Complete the proof of Theorem 0.7 by showing that ~is an equivalence relation on S.Prove that 3, 5, and 7 are the only three consecutive odd integers that are prime.What is the last digit of 3100 ? What is the last digit of 2100 ?Prove that there are no rational numbers x and y such that x2y21002 .(Cancellation Property) Suppose , and are functions. If =and is one-to-one and onto, prove that = .1E2EIn D4 , find all elements X such that a. X3=V ; b. X3=R90 ; c. X3=R0 ; d. X2=R0 ; e. X2=H .4EFor n3 , describe the elements of Dn . (Hint: You will need toconsider two cases—n even and nodd.) How many elements does Dn have?In Dn , explain geometrically why a reflection followed by a reflection must be a rotation.7E8EAssociate the number 1 with a rotation and the number -1 with a reflection. Describe an analogy between multiplying these two numbers and multiplying elements of Dn .If r1,r2,andr3 represent rotations from Dn and f1,f2,andf3 represent reflections from Dn , determine whether r1r2f1r3f2f3r3 is a rotation or a reflection.Suppose that a, b, and c are elements of a dihedral group. Is a2b4ac5a3c a rotation or a reflection? Explain your reasoning.12EFind elements A, B, and C in D4 such that AB=BC but AC.(Thus, “cross cancellation” is not valid.)Explain what the following diagram proves about the group Dn .15EDescribe the symmetries of a parallelogram that is neither a rectangle nor a rhombus. Describe the symmetries of a rhombus that isnot a rectangle.Describe the symmetries of a noncircular ellipse. Do the same for a hyperbola.18E19EDetermine the symmetry group of the outer shell of the cross section of the human immunodeficiency virus (HIV) shown below.Let X,Y,R90 be elements of D4 with YR90andX2Y=R90 . DetermineY. Show your reasoning.If F is a reflection in the dihedral group Dn find all elements X in Dn such that X2=F and all elements X in Dn such that X3=F .What symmetry property do the words “mow,” “sis,” and “swims”have when written in uppercase letters?For each design below, determine the symmetry group (ignore imperfections).What group theoretic property do uppercase letters F, G, J, L, P, Q, R have that is not shared by the remaining uppercase letters in the alphabet?Which of the following binary operations are closed? a. subtraction of positive integers b. division of nonzero integers c. function composition of polynomials with real coefficients d. multiplication of 22 matrices with integer entries e. exponentiation of integersWhich of the following binary operations are associative? a. subtraction of integers b. division of nonzero rationals c. function composition of polynomials with real coefficients d. multiplication of 22 matrices with integer entries e. exponentiation of integersWhich of the following binary operations are commutative? a. substraction of integers b. division of nonzero real numbers c. function composition of polynomials with real coefficients d. multiplication of 22 matrices with real entries e. exponentiation of integersWhich of the following sets are closed under the given operation? a. {0, 4, 8, 12} addition mod 16 b. {0, 4, 8, 12} addition mod 15 c. {1, 4, 7, 13} multiplication mod 15 d. {1, 4, 5, 7} multiplication mod 9In each case, find the inverse of the element under the given operation. a. 13 in Z20 b. 13 in U(14) c. n1 in U(n) (n2) d. 32i in C*, the group of nonzero complex numbers under multiplicationIn each case, perform the indicated operation. a. In C*, (7+5i)(3+2i) b. In GL(2,Z13) , det [7415] c. InGL(2,R)[ 6 3 8 2]1 d. InGL(2,Z7)[ 2 1 1 3]17EList the elements of U(20).Show that {1, 2, 3} under multiplication modulo 4 is not a group but that {1, 2, 3, 4} under multiplication modulo 5 is a group.Show that the group GL(2,R) of Example 9 is non-Abelian by exhibitinga pair of matrices A and B in GL(2,R) such that ABBA .Let a belong to a group and a12=e . Express the inverse of each of the elements a, a6,a8,anda11 in the form ak for some positive integer k.In U(9)find the inverse of 2, 7, and 8.Translate each of the following multiplicative expressions into itsadditive counterpart. Assume that the operation is commutative. a. a2b3 b. a2(b1c)2 c. (ab2)3c2=eFor group elements a, b, and c, express (ab)3and(ab2c)2 withoutparentheses.Suppose that a and b belong to a group and a5=eandb7=e .Write a2b4and(a2b4)2 without using negative exponents.Show that the set {5, 15, 25, 35} is a group under multiplication modulo 40. What is the identity element of this group? Can you seeany relationship between this group and U(8)?Let G be a group and let H=x1xG . Show that G=H as sets.List the members of K=x2xD4andL=xD4x2=e .Prove that the set of all 22 matrices with entries from R and determinant +1 is a group under matrix multiplication.For any integer n2 , show that there are at least two elements inU(n) that satisfy x2=1 .An abstract algebra teacher intended to give a typist a list of nine integers that form a group under multiplication modulo 91. Instead, one of the nine integers was inadvertently left out, so that the list appearedas 1, 9, 16, 22, 53, 74, 79, 81. Which integer was left out?(This really happened!)Let G be a group with the property that for any x, y, z in the group, xy=zx implies y=z . Prove that G is Abelian. (“Left-right cancellation”implies commutativity.)(Law of Exponents for Abelian Groups) Let a and b be elements of an Abelian group and let n be any integer. Show that (ab)n=anbn. Is this also true for non-A belian groups?(SocksShoes Property) Draw an analogy between the statement (ab)1=b1a1 and the act of putting on and taking off your socks and shoes. Find distinct nonidentity elements a and b from a non-Abelian group such that (ab)1=b1a1 . Find an example that shows that in a group, it is possible to have (ab)2b2a2 .What would be an appropriate name for the group property (abc)1=c1b1a1 ?Prove that a group G is Abelian if and only if (ab)1=a1b1 for all a and b in G.Prove that in a group, (a1)1=a for all a.For any elements a and b from a group and any integer n, prove that (a1ba)n=a1bna .If a1,a2,...,an belong to a group, what is the inverse of a1,a2,...,an ?The integers 5 and 15 are among a collection of 12 integers that form a group under multiplication modulo 56. List all 12.30E31EConstruct a Cayley table for U(12).Suppose the table below is a group table. Fill in the blank entries.Prove that in a group, (ab)2=a2b2 if and only if ab=ba . Prove that in a group, (ab)2=b2a2 if and only if ab=ba .Let a, b, and c be elements of a group. Solve the equation axb=c for x. Solve a1xa=c for x.Let a and b belong to a group G. Find an x in G such that xabx1=ba.Let G be a finite group. Show that the number of elements x of Gsuch that x3=e is odd. Show that the number of elements x of G such that x2e is even.Give an example of a group with elements a, b, c, d, and x such that axb=cxd but abcd . (Hence “middle cancellation” is notvalid in groups.)Suppose that G is a group with the property that for every choice of elements in G, axb=cxd implies ab=cd . Prove that G is Abelian. (“Middle cancellation” implies commutativity.)Find an element X in D4 such that R90VXH=D .Suppose F1andF2 are distinct reflections in a dihedral group Dn .Prove that F1F2R0 .Suppose F1andF2 are distinct reflections in a dihedral group Dn such that F1F2=F2F1. Prove that F1F2R180 .Let R be any fixed rotation and F any fixed reflection in a dihedral group. Prove that RkFRk=F .Let R be any fixed rotation and F any fixed reflection in a dihedral group. Prove that FRkF=Rk . Why does this imply that Dn is non-Abelian?In the dihedral group Dn , let R=R360/n and let F be any reflection.Write each of the following products in the form RiorRiF, where 0in . a. In D4,FR2FR5 b. In D5,R3FR4FR2 c. In D6,FR5FR2FProve that the set of all 33 matrices with real entries of the form [ 1 a b 0 1 c 0 0 1] is a group. (Multiplication is defined by [ 1 a b 0 1 c 0 0 1][ 1 a b 0 1 c 0 0 1]=[1a+ab+ac+b01c+c001] . This group, sometimes called the Heisenberg group after the NobelPrizewinning physicist Werner Heisenberg, is intimately related to the Heisenberg Uncertainty Principle of quantum physics.)Prove that if G is a group with the property that the square of every element is the identity, then G is Abelian. (This exercise is referred to in Chapter 26.)In a finite group, show that the number of nonidentity elements that satisfy the equation x5=e is a multiple of 5. If the stipulation that the group be finite is omitted, what can you say about the number of nonidentity elements that satisfy the equation x5=e ?List the six elements of GL(2,Z2) . Show that this group is non-Abelian by finding two elements that do not commute. (This exerciseis referred to in Chapter 7.)Prove the assertion made in Example 19 that the set 1,2,...,n1 is a group under multiplication modulo n if and only if n isprime.Suppose that in the definition of a group G, the condition that there exists an element e with the property ae=ea=a for all ain G is replaced by ae=a for all a in G. Show that ea=a for all a in G.(Thus, a one-sided identity is a two-sided identity.)Suppose that in the definition of a group G, the condition that for each element a in G there exists an element b in G with the property ab=ba=e is replaced by the condition ab=e . Show that ba=e . (Thus, a one-sided inverse is a two-sided inverse.)For each group in the following list, find the order of the group and the order of each element in the group. What relation do you see between the orders of the elements of a group and the order of the group? Z12 , U(10), U(12), U(20), D4Let Q be the group of rational numbers under addition and let Q*be the group of nonzero rational numbers under multiplication.In Q, list the elements in 12 . In Q*, list the elements in 12 .Let Q and Q* be as in Exercise 2. Find the order of each element in Q and in Q*.Prove that in any group, an element and its inverse have the same order.Without actually computing the orders, explain why the two elementsin each of the following pairs of elements from Z30 must havethe same order: {2, 28}, {8, 22}. Do the same for the following pairs of elements from U(15):{2,8},{7,13} .In the group Z12 , find a,b,anda+b for each case. a. a=6,b=2 b. a=3,b=8 c. a=5,b=4 Do you see any relationship between a,b,anda+b ?If a, b, and c are group elements and a=6,b=7 , express (a4c2b4)1 without using negative exponents.What can you say about a subgroup of D3 that contains R240 and a reflection F? What can you say about a subgroup of D3 that contains two reflections?What can you say about a subgroup of D4 that contains R270 and a reflection? What can you say about a subgroup of D4 that contains H and D? What can you say about a subgroup of D4 that contains H and V?How many subgroups of order 4 does D4 have?Determine all elements of finite order in R*, the group of nonzeroreal numbers under multiplication.Complete the statement “A group element x is its own inverse if and only if x= ________.”For any group elements a and x, prove that xax1=a . This exercise is referred to in Chapter 24.Prove that if a is the only element of order 2 in a group, then a lies in the center of the group.(1969 Putnam Competition) Prove that no group is the union of two proper subgroups. Does the statement remain true if “two” is replaced by “three”?Let G be the group of symmetries of a circle and R be a rotation of the circle of 2 degrees. What is R ?For each divisor k1 of n, let Uk(n)=xU(n)xmodk=1 .[For example, U3(21)={1,4,10,13,16,19} and U7(21)={1,8} .]List the elements of U4(20),U5(20),U5(30),andU10(30) . Prove that Uk(n) is a subgroup of U(n). Let H=xU(10)xmod3=1 . Is H a subgroup of U(10)? (This exercise is referred to in Chapter 8.)Suppose that a is a group element and a6=e . What are the possibilities for a ? Provide reasons for your answer.If a is a group element and a has infinite order, prove that amanwhenmn .For any group elements a and b, prove that ab=ba .Show that if a is an element of a group G, then a||G.Show that U(14)=3=5 . [Hence, U(14) is cyclic.] Is U(14)=11 ?Show that U(20)k for any k in U(20). [Hence, U(20) is notcyclic.]Suppose n is an even positive integer and H is a subgroup of Zn .Prove that either every member of H is even or exactly half of the members of H are even.Let n be a positive even integer and let H be a subgroup of Zn of oddorder. Prove that every member of H is an even integer.Prove that for every subgroup of Dn , either every member of the subgroup is a rotation or exactly half of the members are rotations.Let H be a subgroup of Dn of odd order. Prove that every member of H is a rotation.Prove that a group with two elements of order 2 that commute musthave a subgroup of order 4.29E30E31ESuppose that H is a subgroup of Z under addition and that H contains 250and350 . What are the possibilities for H?Prove that the dihedral group of order 6 does not have a subgroup of order 4.If H and K are subgroups of G, show that HK is a subgroup of G.(Can you see that the same proof shows that the intersection of any number of subgroups of G, finite or infinite, is again a subgroup of G?)Let G be a group. Show that Z(G)=aGC(a) . [This means the intersection of all subgroups of the form C(a).]Let G be a group, and let aG . Prove that C(a)=C(a1) .For any group element a and any integer k, show that C(a)C(ak) .Use this fact to complete the following statement: “In a group, if x commutes with a, then . . . .” Is the converse true?Let G be an Abelian group and H=xG||x is odd}. Prove that H is a subgroup of G.39E40ELet Sbe a subset of a group and let H be the intersection of all subgroups of G that contain S. a. Prove that S=H . b. If S is nonempty, prove that S=s1n1s2n2smnmm1,siS,niZ . (The siterms need not be distinct.)In the group Z, find a. 8,14 ; b. 8,13 ; c. 6,15 ; d. m,n ; e. 12,18,45 . In each part, find an integer k such that the subgroup is k .Prove Theorem 3.6. Theorem 3.6 C(a) Is a Subgroup For each a in a group G, the centralizer of a is a subgroup of G. PROOF A proof similar to that of Theorem 3.5 is left to the reader to supply (Exercise 43). Notice that for every element a of a group G,Z(G)C(a) . Also, observe that G is Abelian if and only if C(a)=G for all ain G.If H is a subgroup of G, then by the centralizer C(H) of H we meanthe set xGxh=hx for all hH . Prove that C(H) is a subgroup of G.Must the centralizer of an element of a group be Abelian? Must the center of a group be Abelian?Suppose a belongs to a group and a=5 . Prove that C(a)=C(a3) .Find an element a from some group such that a=6andC(a)C(a3) .47EIn each case, find elements a and b from a group such that a|=|b=2 . a. ab=3 b. ab=4 c. ab=5 Can you see any relationship among a,b,andab ?Prove that a group of even order must have an odd number of elements of order 2.Consider the elements A=[0110]andB=[0111] from SL(2,R) . Find A|,|B|,and|AB . Does your answer surprise you?51EGive an example of elements a and b from a group such that a has finite order, b has infinite order and ab has finite order.Consider the element A=[1101] in SL(2,R) . What is the order of A? If we view A=[1101] as a member of SL(2,Zp) (p is a prime),what is the order of A?For any positive integer n and any angle , show that in the group SL(2,R) , [cossinsincos]=[cosnsinnsinncosn] .Use this formula to find the order of [cos60sin60sin60cos60]=[cos2sin2sin2cos2] .(Geometrically, [cossinsincos] represents a rotation of the plane degrees.)55EIn the group R* find elements a and b such that a=,b=andab=2 .57E58E59ECompute the orders of the following groups. a. U(3),U(4),U(12) b. U(5),U(7),U(35) c. U(4),U(5),U(20) d. U(3),U(5),U(15) On the basis of your answers, make a conjecture about the relationship among U(r),U(s),andU(rs) .Let R* be the group of nonzero real numbers under multiplication and let H=xRx2 is rational}. Prove that H is a subgroup of R*. Can the exponent 2 be replaced by any positive integer and still have H be a subgroup?Compute U(4),U(10),andU(40) . Do these groups provide acounter example to your answer to Exercise 60? If so, revise your conjecture.Find a noncyclic subgroup of order 4 in U(40).Prove that a group of even order must have an element of order 2.Let G={[abcd]|a,b,c,dZ} under addition. Let H={[abcd]|G,a+b+c+d=0} . Prove that H is a subgroup of G.What if 0 is replaced by 1?Let H=AGL(2,R)detA is an integer power of 2}. Show that H is a subgroup of GL(2,R) .Let H be a subgroup of R under addition. Let K=2aaH .Prove that K is a subgroup of R* under multiplication.Let G be a group of functions from R to R*, where the operation of G is multiplication of functions. Let H=fGf(2)=1 . Prove that H is a subgroup of G. Can 2 be replaced by any real number?Let G=GL(2,R) and H={[a00b]|aandbarenonzerointegers} under the operation of matrix multiplication. Prove ordisprove that H is a subgroup of GL(2,R) .Let H=a+bia,bR,ab0 . Prove or disprove that H is asubgroup of C under addition.Let H=a+bia,bR,a2+b2=1 . Prove or disprove that H is a subgroup of C* under multiplication. Describe the elements of H geometrically.Let G be a finite Abelian group and let a and b belong to G. Prove that the set a,b=aibji,jZ is a subgroup of G. What can you say about a,b in terms of aandb ?73EIf H and K are nontrivial subgroups of the rational numbers underaddition, prove that HK is nontrivial.75EProve that a group of order n greater than 2 cannot have a subgroup of order n1 .Let a belong to a group and a=m. If n is relatively prime to m,show that a can be written as the nth power of some element in the group.Let G be a finite group with more than one element. Show that G has an element of prime order.Find all generators of Z6,Z8,andZ20 .Suppose that a,b,andc are cyclic groups of orders 6, 8, and20, respectively. Find all generators of a,b,andc .List the elements of the subgroups 20and10inZ30 . Let a be agroup element of order 30. List the elements of the subgroups a20anda10 .List the elements of the subgroups 3and15inZ18 . Let a be agroup element of order 18. List the elements of the subgroups a3anda15 .List the elements of the subgroups 3and7inU(20) .What do Exercises 3, 4, and 5 have in common? Try to make a generalization that includes these three cases.Find an example of a noncyclic group, all of whose proper subgroupsare cyclic.Let a be an element of a group and let a=15 . Compute the ordersof the following elements of G. a. a3,a6,a9,a12 b. a5,a10 c. a2,a4,a8,a149EIn Z24 , list all generators for the subgroup of order 8. Let G=aandleta=24 . List all generators for the subgroup of order 8.Let G be a group and let aG . Prove that a1=a .In Z, find all generators of the subgroup 3 . If a has infinite order,find all generators of the subgroup a3 .In Z24 , find a generator for 2110 . Suppose that a=24 . Finda generator for a21a10 . In general, what is a generator for thesubgroup aman ?Suppose that a cyclic group G has exactly three subgroups: G itself,{e}, and a subgroup of order 7. What is |G|? What can you say if 7is replaced with p where p is a prime?Let G be an Abelian group and let H=gG||g divides 12}.Prove that H is a subgroup of G. Is there anything special about 12here? Would your proof be valid if 12 were replaced by some otherpositive integer? State the general result.Complete the statement: a|=|a2 if and only if |a| . . . .Complete the statement: a2|=|a12 if and only if . . . .Let a be a group element and a= . Complete the followingstatement: ai=aj if and only if . . . .If a cyclic group has an element of infinite order, how many elementsof finite order does it have?Suppose that G is an Abelian group of order 35 and every elementof G satisfies the equation x35=e . Prove that G is cyclic. Does yourargument work if 35 is replaced with 33?Let G be a group and let a be an element of G. a. If a12=e , what can we say about the order of a? b. If am=e , what can we say about the order of a? c. Suppose that G=24 and that G is cyclic. If a8eanda12e ,show that a=G .Prove that a group of order 3 must be cyclic.Let Z denote the group of integers under addition. Is every subgroupof Z cyclic? Why? Describe all the subgroups of Z. Let a be a groupelement with infinite order. Describe all subgroups of a .For any element a in any group G, prove that a is a subgroup ofC(a) (the centralizer of a).If d is a positive integer, d2 , and d divides n, show that the numberof elements of order d in Dn is (d) . How many elements oforder 2 does Dn have?Find all generators of Z. Let a be a group element that has infiniteorder. Find all generators of a .Prove that C*, the group of nonzero complex numbers under multiplication,has a cyclic subgroup of order n for every positive integer n.Let a be a group element that has infinite order. Prove that ai=aj if and only if i=j.List all the elements of order 8 in Z8000000 . How do you know your listis complete? Let a be a group element such that a=8000000 . Listall elements of order 8 in a . How do you know your list is complete?Suppose that G is a group with more than one element. If the onlysubgroups of G are {e} and G, prove that G is cyclic and has primeorder.Let G be a finite group. Show that there exists a fixed positive integern such that an=e for all ain G. (Note that n is independent of a.)Determine the subgroup lattice for Z12 . Generalize to Zp2q , where pand q are distinct primes.Determine the subgroup lattice for Z8 . Generalize to Zpn , where p isa prime and n is some positive integer.Prove that a finite group is the union of proper subgroups if andonly if the group is not cyclic.Show that the group of positive rational numbers under multiplicationis not cyclic. Why does this prove that the group of nonzerorationals under multiplication is not cyclic?Consider the set {4, 8, 12, 16}. Show that this set is a group undermultiplication modulo 20 by constructing its Cayley table. Whatis the identity element? Is the group cyclic? If so, find all of its generators.Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group itself). Generalize to exactly nsubgroups for any positive integer n.Let m and n be elements of the group Z. Find a generator for thegroup mn .Suppose that a andb are group elements that commute. If |a| is Finiteand |b| infinite, prove that |ab| has infinite order.40E41ELet F and F’be distinct reflections in D21 . What are the possibilitiesfor |FF’|?Suppose that H is a subgroup of a group G and H=10 . If abelongs to G and a6 belongs to H, what are the possibilities for |a|?44EIf G is an infinite group, what can you say about the number ofelements of order 8 in the group? Generalize.If G is a cyclic group of order n, prove that for every element a in G, an=e .For each positive integer n, prove that C*, the group of nonzerocomplex numbers under multiplication, has exactly (n) elementsof order n.Prove or disprove that H=nZn is divisible by both 8 and 10}is a subgroup of Z. What happens if “divisible by both 8 and 10” ischanged to “divisible by 8 or 10?”49E50E51E52E53E54E55E56E57E58EProve that no group can have exactly two elements of order 2.Given the fact that U(49) is cyclic and has 42 elements, deduce thenumber of generators that U(49) has without actually finding any ofthe generators.Let a andb be elements of a group. If a=10andb=21 , showthat ab={e} .Let a andb belong to a group. If |a| and |b| are relatively prime,show that ab={e} .Let a andb belong to a group. If a=24andb=10 , what are thepossibilities for ab ?Prove that U(2n)(n3) is not cyclic.Prove that for any prime p and positive integer n,(pn)=pnpn1 .Prove that Zn has an even number of generators if n2 . What doesthis tell you about (n) ?If a5=12 , what are the possibilities for |a|? If a4=12 , what arethe possibilities for |a|?Suppose that x=n . Find a necessary and sufficient condition on rand s such that xrkxs .Let a be a group element such that a=48 . For each part, find adivisor k of 48 such that a21=ak; a14=ak; a18=ak .Prove that H={[1n01]|nZ} is a cyclic subgroup of GL(2,R) .Suppose that |a| and |b| are elements of a group and a andb commute.If a=5andb=16 , prove that ab=80 .Let a andb belong to a group. If a=12,b=22,andabe , prove that a6=b11 .Determine (81),(60)and(105) where is the Euler phifunction.If n is an even integer prove that (2n)=2(n) .Let a andb belong to some group. Suppose that a=m,b=n ,and m and n are relatively prime. If ak=bk for some integer k,prove that mndivides k. Give an example to show that the conditionthat m and n are relatively prime is necessary.For every integer n greater than 2, prove that the group U(n21) is not cyclic.(2008 GRE Practice Exam) If x is an element of a cyclic group oforder 15 and exactly two of x3,x5,andx9 are equal, determine x13 .Let [123456213546]and=[123456612435] . Compute each of the following. a. 1 b. c.Let [1234567823451786]and=[1234567813876524] . Write , , and as a. products of disjoint cycles; b. products of 2-cycles.Write each of the following permutations as a product of disjointcycles. a. (1235)(413) b. (13256)(23)(46512) c. (12)(13)(23)(142)