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MATH 4441 & MATH 6441: MODERN ALGEBRA I
HOMEWORK SETS AND EXAMS
Yongwei Yao
2009 SPRING SEMESTER GEORGIA STATE UNIVERSITY ??
Contents HW Set #01, Problems 1 HW Set #02, Problems 2 HW Set #03, Problems 3 HW Set #04, Problems 4 HW Set #05, Problems 5 Midterm I, Review 6 Midterm I, Problems 7 HW Set #06, Problems 8 HW Set #07, Problems 9 HW Set #08, Problems 10 HW Set #09, Problems 11 Midterm II, Review 12 Midterm II, Problems 13 HW Set #10, Problems 14 HW Set #11, Problems 15 HW Set #12, Problems 16 Final Exam, Review 17 Final Exam, Problems 18 Extra Credit Set, Problems 19
Note. There are four (4) problems in each homework set. Math 6441 students need to do all 4 problems while Math 4441 students need to do any three (3) problems out the four. If a Math 4441 student submits all 4 problems, then one of the lowest score(s) is dropped. There is a bonus point for a Math 4441 student doing all 4 problems correctly. There are three (3) PDF files for the homework sets and exams, one with the problems only, one with hints, and one with solutions. Links are available below.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #01 (Due 01/13) Problems
Problem 1.1. Let A = {a, b, c, d}, B = {b, d, e, f, g} and C = {f, g, h} be sets.
(1) Determine A ∩ B and B ∪ C. (2) Determine (A ∩ B) ∪ C and A ∩ (B ∪ C). Are they equal? (3) Determine A \ B and B ∩ C. (Note that A \ B may be also denoted by A − B.) (4) Determine (A \ B) ∩ C and A \ (B ∩ C). Are they equal?
Problem 1.2. Let A, B and C be sets.
(1) Show A \ (B ∩ C) = (A \ B) ∪ (A \ C). (2) Show A \ (B ∪ C) = (A \ B) \ C.
Problem 1.3. For each function fi, determine whether it is injective but not surjective, surjective but not injective, bijective, or neither injective nor surjective. Justify.
(1) f 1 : R → R≥ 0 with f 1 (x) = x^2 for all x ∈ R, in which R≥ 0 = {x ∈ R | x ≥ 0 } = [0, ∞). (2) f 2 : R≥ 0 → R≥ 0 with f 2 (x) = x^2 for all x ∈ R≥ 0. (3) f 3 : R → R with f 3 (x) = 10x^ for all x ∈ R. (4) f 4 : R → R with f 4 (x) = 10x 2 for all x ∈ R. (Note 10x 2 stands for 10(x (^2) ) , not (10x)^2 .)
Problem 1.4 (Math 6441). Let A, B and C be sets.
(1) Show A ∪ (B \ C) ⊇ (A ∪ B) \ C. (2) Find explicit examples of A, B and C such that A ∪ (B \ C) ) (A ∪ B) \ C.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #03 (Due 01/27) Problems
Problem 3.1. Determine whether the following statement is true or false with justification: If f : X → Y and g : Y → X are functions such that g ◦ f = IX , then f ◦ g = IY.
Problem 3.2. Let S 3 denote the set of all bijective functions from X = { 1 , 2 , 3 } to itself. Let ϕ ∈ S 3 and ψ ∈ S 3 be defined as follows
ϕ : 1 7 → 2 , 2 7 → 3 , 3 7 → 1 and ψ : 1 7 → 3 , 2 7 → 2 , 3 7 → 1 (1) Determine ϕ ◦ ψ and ψ ◦ ϕ explicitly. Are they equal? (2) Determine ϕ−^1 and ψ−^1 explicitly. (3) Determine ϕ^2 and ϕ^3 explicitly. Is anyone of the two equal to IX? (4) Determine ψ^2 and ψ^3 explicitly. Is anyone of the two equal to IX?
Problem 3.3. Consider integers 24, 60, 26 and 46.
(1) List all (positive and negative) common divisors of 24 and 60. Determine gcd(24, 60). (2) Express gcd(26, 46) as a linear combination of 26 and 46 (with integer coefficients).
Problem 3.4 (Math 6441). Let f : X → Y and g : Y → X be functions satisfying g◦f = IX. Show that f ◦ g = IY if and only if f is onto.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #04 (Due 02/03) Problems
Problem 4.1. Determine whether each of the statements is true or false with justification.
(1) There exists a sequence of 13 consecutive positive integers less than 100 that contains exactly 2 primes. (2) There are at least 2 primes in every sequence of 7 consecutive positive integers. (3) If m and n are natural numbers, then (mn)! = m! n!.
Problem 4.2. Let x = 1 + 4i, y = 2 − 3 i and z = −
3 + 3i. (1) Compute x + y, x − y, xy and x/y. (2) Write z in polar form z = r(cos θ + i sin θ) with 0 ≤ r ∈ R and 0 ≤ θ < 2 π. (3) Compute z^6. Is z^6 a real number? Show your reasoning/computation.
Problem 4.3. For all m, n ∈ Z, let m ∗ n = m + n, the sum of m and n. For each of the statements, determine whether it is true or false with justification.
(1) For all a, b ∈ Z, one has a ∗ b ∈ Z. (2) For all a, b, c ∈ Z, one has (a ∗ b) ∗ c = a ∗ (b ∗ c). (3) There exists a (fixed) element e ∈ Z such that e ∗ a = a ∗ e = a for all a ∈ Z. (4) For every a ∈ Z, there exists a′^ ∈ Z such that a ∗ a′^ = a′^ ∗ a = e.
Problem 4.4 (Math 6441). For all m, n ∈ Z, let m ∗ n = mn, the product of m and n. For each of the statements, determine whether it is true or false with justification.
(1) a ∗ b ∈ Z for all a, b ∈ Z. (2) (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ Z. (3) There exists a (fixed) element e ∈ Z such that e ∗ a = a ∗ e = a for all a ∈ Z. (4) For every a ∈ Z, there exists a′^ ∈ Z such that a ∗ a′^ = a′^ ∗ a = e.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Midterm Exam I (02/17/2009) Review
Functions: Problems 2.1, 2.2, 2.3, 2.4, 3.1, 3.4.
About S 3 : Problem 3.2.
Complex numbers: Problem 4.2.
Definition of groups: Problems 4.3, 4.4, 5.1, 5.2, 5.3, 5.4.
Lecture notes and textbooks: All we have covered.
Note: The above list is not intended to be complete. The problems in
the actual test may vary in difficulty as well as in content. Going over,
understanding, and digesting the problems listed above will definitely help.
However, simply memorizing the solutions of the problems may not help you
very much.
You are strongly encouraged to practice more problems (than the ones
listed above) on your own.
Click here for the actual Midterm Exam I.
Math 4441/6441 (Spring 2009) Midterm Exam I (02/17/2009) Problems
Problem I.1 (5 points). Let x = 5 + 4i, y = −3 + 2i and z = −
3 i. (1) Compute xy and x/y. (2) Write z in polar form z = r(cos θ + i sin θ) with 0 ≤ r ∈ R and 0 ≤ θ < 2 π. (3) Compute z^30 and express it in terms of z^30 = a + bi with a, b ∈ R.
Problem I.2 (5 points). Consider S 3 , which consists of six (6) bijective functions below
f 1 : 1 7 → 1 , 2 7 → 2 , 3 7 → 3; f 2 : 1 7 → 1 , 2 7 → 3 , 3 7 → 2; f 3 : 1 7 → 2 , 2 7 → 1 , 3 7 → 3; f 4 : 1 7 → 2 , 2 7 → 3 , 3 7 → 1; f 5 : 1 7 → 3 , 2 7 → 1 , 3 7 → 2; f 6 : 1 7 → 3 , 2 7 → 2 , 3 7 → 1.
Fill in each of the blanks with (one of) f 1 , f 2 , f 3 , f 4 , f 5 , f 6. (1 point each.)
f 2 ◦ f 3 = , f 3 ◦ f 2 = , f 2 − 1 = , f 5 − 1 = , f 2 − 1 ◦ f 5 − 1 =.
Problem I.3 (5 points). Let f : X → Y and g : Y → Z be functions, in which X, Y and Z are non-empty sets. Prove or disprove each of the following statements.
(1) If f is injective, then g ◦ f is injective. (2) If g is onto, then g ◦ f is onto.
Problem I.4 (5 points). Let X = {x ∈ R | 0. 01 < x < 90 }, an open interval in R. For all x, y ∈ X, define x ∗ y = xy, the usual multiplication. Determine whether each of the statements (1)–(4) is true or false with justification. Then complete (5).
(1) x ∗ y ∈ X for all x, y ∈ X. (2) (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x, y, z ∈ X. (3) There exists a (fixed) element e ∈ X such that e ∗ x = x ∗ e = x for all x ∈ X. (4) For every x ∈ X, there exists x′^ ∈ X such that x′^ ∗ x = x ∗ x′^ = e. (5) (X, ∗) is an abelian group a non-abelian group not a group (choose one)
Problem I.5 (5 points). Let X = {a + bi | 0 ≤ a ∈ R, b ∈ R} and Y = X \ { 0 }, both in C.
(1) Determine whether (X, +) is a group, where + denotes addition. Show work. (2) Determine whether (Y, · ) is a group, where · denotes multiplication. Show work.
Extra Credit Problem I.6 (1 point, no partial credit). Let f : X → Y and g : Y → X be functions satisfying g ◦ f = IX. If g is injective, show f ◦ g = IY.
Extra Credit Problem I.7 (1 point, no partial credit). Let (G, ∗) be a group of order 3, i.e., |G| = 3. Show (G, ∗) is abelian.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #07 (Due 03/10) Problems
Problem 7.1. Consider the group (S 3 , ◦), which consists of six (6) bijective functions below
f 1 : 1 7 → 1 , 2 7 → 2 , 3 7 → 3; f 2 : 1 7 → 1 , 2 7 → 3 , 3 7 → 2; f 3 : 1 7 → 2 , 2 7 → 1 , 3 7 → 3; f 4 : 1 7 → 2 , 2 7 → 3 , 3 7 → 1; f 5 : 1 7 → 3 , 2 7 → 1 , 3 7 → 2; f 6 : 1 7 → 3 , 2 7 → 2 , 3 7 → 1.
Fill in each blank in (1) and (2) with (one of) f 1 , f 2 , f 3 , f 4 , f 5 , f 6. Complete (3) and (4).
(1) f 5 ◦ f 6 = , (f 5 ◦ f 6 )−^1 =. (2) f 5 − 1 = , f 6 − 1 = , f 5 − 1 ◦ f 6 − 1 = , f 6 − 1 ◦ f 5 − 1 =. (3) Does the equation (f 5 ◦ f 6 )−^1 = f 5 − 1 ◦ f 6 − 1 hold? How about (f 5 ◦ f 6 )−^1 = f 6 − 1 ◦ f 5 − 1? (4) Find o(f 1 ), o(f 3 ) and o(f 4 ), the order of the elements.
Problem 7.2. Let G be a group and let a, b be (fixed) elements of G.
(1) If a−^1 = b−^1 , show a = b. (2) True or false: If a 6 = b, then a−^1 6 = b−^1. Justify your claim. (3) Assume ab = ba. Show (a) ab−^1 = b−^1 a; (b) a−^1 b = ba−^1 ; (c) a−^1 b−^1 = b−^1 a−^1.
Problem 7.3. Let G be an abelian group and let N = {g ∈ G | o(g) < ∞}.
(1) True or false: (ab)n^ = anbn^ for all a, b ∈ G and all n ∈ Z. No need to justify. (2) True or false: e ∈ N. Justify your claim. (3) For all x, y ∈ N , show (a) xy ∈ N ; (b) x−^1 ∈ N.
Problem 7.4 (Math 6441). Let G be a group such that (ab)^4 = a^4 b^4 , (ab)^5 = a^5 b^5 and (ab)^6 = a^6 b^6 for all a, b ∈ G. Show G is abelian. (Compare this with Problem 8.4.)
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #08 (Due 03/17) Problems
Problem 8.1. Let A = C \ { 0 }. It is known (and easy to prove) that (A, · ) is a group under the usual multiplication while (C, +) is a group under the usual addition.
(1) Determine the orders of 1, i, −^12 +
√ 3 2 i^ and 2, considered as elements of (A,^ ·^ ). (2) Determine the orders of 0, 1 and i, if they are considered as elements of (C, +).
Problem 8.2. Let G be an abelian group and n a (fixed) integer. Let H = {x ∈ G | xn^ = e}.
(1) Show e ∈ H. (Thus H 6 = ∅.) (2) For all x, y ∈ H, show xy ∈ H. (3) For all x ∈ H, show x−^1 ∈ H.
Problem 8.3. Let G be a group and let g ∈ G (be a fixed element). Define h : G → G by h(x) = gxg−^1 for all x ∈ G. Determine whether each of the following statements is true or false with justification.
(1) h(x) = x if xg = gx. (2) h(x) = x only if xg = gx. (Or, worded differently, xg = gx if h(x) = x.) (3) h(xy) = h(x)h(y) for all x, y ∈ G. (4) h(x−^1 ) = (h(x))−^1 for all x ∈ G.
Problem 8.4 (Math 6441). Let G be a group. Assume that, for some (fixed) integer n ≥ 1, we have (ab)n^ = anbn, (ab)n+1^ = an+1bn+1^ and (ab)n+2^ = an+2bn+2^ for all a, b ∈ G. Show G is abelian. (Compare this with Problem 7.4.)
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Midterm Exam II (03/31/2009) Review
Explicit computations, orders of elements: Problems 7.1, 7.3, 8.1, 9.1, 9.2.
Basic properties of groups: Problems 7.2, 7.3, 8.2, 8.3,....
Proving a group is abelian: Problems 6.2, 6.3, 7.4, 8.4.
Homomorphisms, kernels, images: Problems 8.3, 9.1, 9.2, 9.3.
Subgroups: Problems 7.3, 8.2, 9.3, 9.4.
Lecture notes and textbooks: All we have covered.
Note: The above list is not intended to be complete. The problems in
the actual test may vary in difficulty as well as in content. Going over,
understanding, and digesting the problems listed above will definitely help.
However, simply memorizing the solutions of the problems may not help you
very much.
You are strongly encouraged to practice more problems (than the ones
listed above) on your own.
Click here for the actual Midterm Exam II.
Math 4441/6441 (Spring 2009) Midterm Exam II (03/31/2009) Problems
Problem II.1. Consider the group (S 3 , ◦) under composition, which consists of the following
f 1 : 1 7 → 1 , 2 7 → 2 , 3 7 → 3; f 2 : 1 7 → 1 , 2 7 → 3 , 3 7 → 2; f 3 : 1 7 → 2 , 2 7 → 1 , 3 7 → 3; f 4 : 1 7 → 2 , 2 7 → 3 , 3 7 → 1; f 5 : 1 7 → 3 , 2 7 → 1 , 3 7 → 2; f 6 : 1 7 → 3 , 2 7 → 2 , 3 7 → 1.
Define h : (Z, +) → (S 3 , ◦) by h(m) = f 5 m for all m ∈ Z, which is a group homomorphism. Fill in each blank with an integer or fi with 1 ≤ i ≤ 6. Complete (4).
(1) o(f 5 ) = , o(f 6 ) =. (3) h(−8) = , h(11) =.
(2) h(−6) = , h(9) =. (4) List 10 elements in Ker(h).
Problem II.2. Let h : G → G′^ be a group homomorphism (with G and G′^ being groups). If h is surjective and G is abelian, show G′^ is abelian.
Problem II.3. Let h : G → G′^ be a group homomorphism and K < G′. (As always, denote the identity elements of G and G′^ by e and e′^ respectively.) Define H = {x ∈ G | h(x) ∈ K}.
(1) Show H < G. (2) Show Ker(h) ⊆ H.
Problem II.4. Let G be an abelian group and N be a (fixed) subgroup of G. Denote H = {a^28 | a ∈ N } and F = {a^4 | a ∈ N }.
(1) Show H < G. (2) Show H ⊆ F.
Problem II.5. Let G be a group. Define h : G → G by h(x) = x^2 for all x ∈ G.
(1) If h is a group homomorphism, show G is abelian. (2) If G is abelian, show h is a group homomorphism.
Extra Credit Problem II.6 (1 point, no partial credit). Prove or disprove: If h : G → G′ is a group homomorphism and G is abelian, then G′^ is abelian.
Extra Credit Problem II.7 (1 point, no partial credit). Prove or disprove: If h : G → G′ is a group homomorphism and x ∈ G such that o(h(x)) < ∞, then o(x) < ∞.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #11 (Due 04/14) Problems
Problem 11.1. Consider the group S 3 under composition, which consists of the following
f 1 : 1 7 → 1 , 2 7 → 2 , 3 7 → 3; f 2 : 1 7 → 1 , 2 7 → 3 , 3 7 → 2; f 3 : 1 7 → 2 , 2 7 → 1 , 3 7 → 3; f 4 : 1 7 → 2 , 2 7 → 3 , 3 7 → 1; f 5 : 1 7 → 3 , 2 7 → 1 , 3 7 → 2; f 6 : 1 7 → 3 , 2 7 → 2 , 3 7 → 1.
Let H = {f 1 , f 2 }, which is a subgroup of S 3.
(1) Determine whether H is a normal subgroup of (S 3 , ◦). Show your reasoning. (2) Find the least i with 1 ≤ i ≤ 6 such that fiH 6 = Hfi.
Problem 11.2. Let G be a group and Hi < G for i = 1, 2.
(1) Show H 1 ∩ H 2 < G. (2) Further assume Hi G for i = 1, 2. Show H 1 ∩ H 2 G.
Problem 11.3. Consider the group (Z, +). Let H = { 4 n | n ∈ Z}, which is a subgroup of (Z, +). In other words, H = {... , − 8 , − 4 , 0 , 4 , 8 ,... } consisting of all multiples of 4.
(1) Determine the (left) cosets 0 + H, 1 + H, 3 + H and 4 + H explicitly. (2) True or false: (−3) + H = 1 + H. No justification is needed. (3) True or false: (−2) + H = 2 + H. No justification is needed. (4) How many distinct (left) cosets of H are there inside the group (Z, +)? (5) Is Z a disjoint union of the distinct (left) cosets? No justification is needed.
Problem 11.4 (Math 6441). Let G be a group and N G. For every (fixed) x ∈ G, show xN = N x.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Homework Set #12 (Due 04/21) Problems
Problem 12.1. Let G be a group of order 60, i.e., |G| = 60.
(1) If x ∈ G satisfies x^28 = e, determine all the possible values of o(x). (2) If y ∈ G satisfies y^18 = e and y^22 6 = e, determine all the possible values of o(y). (3) Is it ever possible to have an element z ∈ G such that z^16 = e and z^36 6 = e? why?
Problem 12.2. Consider the group (Z, +). Let N = { 8 n | n ∈ Z}, which is a normal subgroup of (Z, +). In other words, N = {... , − 16 , − 8 , 0 , 8 , 16 ,... } consisting of all multiples of 8. Denote by Z 8 the quotient group of (Z, +) by N (whose operation is +).
(1) True or false: Z 8 = {0 + N, 1 + N,... , 7 + N }. No justification is needed. (2) True or false: Z 8 = {1 + N, 2 + N,... , 8 + N }. No justification is needed. (3) Find o(6 + N ). (4) Determine [6 + N ], the cyclic subgroup of Z 8 generated by 6 + N. (5) Is Z 8 a cyclic group? If so, find a generator of Z 8.
Problem 12.3. Let G be an abelian group and let N be a subgroup of G, i.e., N < G.
(1) Show that N G. (2) Show that G/N abelian.
Problem 12.4 (Math 6441). Let G be a group and N G. Consider the quotient group G/N. Assume xyx−^1 y−^1 ∈ N for all x, y ∈ G. Show G/N is abelian.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Final Exam (04/28/2009) Problems
Problem III.1. Consider (Z 12 , +), which is a quotient group of (Z, +) modulo its normal subgroup N = 12Z = { 12 n | n ∈ Z}. Thus Z 12 = {0 + N, 1 + N,... , 11 + N }.
(1) Find o(9 + N ) and o(10 + N ). (2) Determine [9 + N ] and [10 + N ] by listing their distinct elements explicitly.
Problem III.2. Let G be a group of order 40, i.e., |G| = 40.
(1) Is it ever possible to have an element x ∈ G such that x^28 = e and x^12 6 = e? If so, what are the possible values of o(x)? Show your reasoning. (2) Is it ever possible to have an element y ∈ G such that y^30 = e and y^15 6 = e? If so, what are the possible values of o(y)? Show your reasoning.
Problem III.3. Let G be a group, H < G and N < G such that N ⊆ Z(G). (1) Show HN < G. (2) Show N G.
Problem III.4. Let G, H and N be as in Problem III.3 above. (Thus HN < G.) Further assume H = [a] for some (fixed element) a ∈ G. Show HN is abelian.
Problem III.5. Let h : G → G′^ be a group homomorphism (with G and G′^ groups) and M < G. Denote N = {h(m) | m ∈ M }. (Denote the identities of G and G′^ by e and e′.)
(1) Show N < G′. (2) Assume h is onto and M G. Show N G′.
Problem III.6. Let G be an abelian group, N = {x ∈ G | x^6 = e} and a ∈ G (a fixed element) such that o(a) = 4. It is known that N G by Problem 8.2 and Problem 12.3.
(1) Find o(a^2 ) and o(a^3 ). Show work. (2) For aN ∈ G/N , determine o(aN ).
Extra Credit Problem III.7 (1 point, no partial credit). True or false: If G is a group and Hi < G for i = 1, 2, then H 1 ∪ H 2 < G. Give a proof or a counterexample.
Extra Credit Problem III.8 (1 point, no partial credit). Let G be a group, N G and a, b ∈ G such that ab ∈ N. Show ba ∈ N.
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
Math 4441/6441 (Spring 2009) Extra Credit Set Problems
For each problem, you get the extra credit points only if you solve it completely and correctly. You may attempt a problem for as many times as you wish by the deadline, or by April 23, 2009 if no explicit deadline is given. Feel free to use the results we have covered.
The points you get from the extra credit problems will be added to the total points you will receive from the homework assignments.
The numbers of? represent the numbers of correct solutions submitted.
Problem E-1 (3 points). Let f : X → Y and g : Y → X be functions satisfying g ◦ f = IX. Show f ◦ g = IY if and only if g is injective.??
Problem E-2 (3 points, deadline 3/17). Let G be a group of order 4, i.e., |G| = 4. Show G is abelian. Do not use Lagrange’s Theorem.??
Problem E-3 (3 points). Let G be a group satisfying (ab)^3 = a^3 b^3 and (ab)^5 = a^5 b^5 for all a, b ∈ G. Show G is abelian. (Compare this with Problem 8.4.)?
Problem E-4 (3 points). Prove or disprove: o(r(cos θ + i sin θ)) < ∞ if and only if r = 1 and θπ ∈ Q, in which r(cos θ + i sin θ) is a non-zero complex number in polar form considered as an element in the group (C \ { 0 }, · ) under the usual multiplication.?
PROBLEMS HINTS SOLUTIONS
Notice. The quoted results are from the textbook Basic Abstract Algebra (2nd edition) by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press.
