Homework for MATH 3321 in Fall 2023


return to Fall 2023 MATH 3321 webpage

Recommended textbook
(required):		 A First Course in Abstract Algebra, 8th Ed, by J. B. Fraleigh & N. E. Brand,
Pearson, 2021.
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  • Homework will not be collected.
  • There might be more than one correct answer for any given question.
  • Dates indicate homework assigned in lecture on that date. Dates for future assignments are tentative and subject to change.
  • If you notice some questions are in a nonincreasing order, then it means that the order is recommended by your instructor and is usually chosen to match the order in which the material was presented in class.
  • Any optional exercises are for those students who wish to explore some of the ideas in more depth – those questions are NOT game for the tests.
  • Skimming through the main ideas in a section shortly before that section is covered in class should help you understand the lecture – try it!
  • The homework assignments below are essentially the assignments found here from Fall 2019, but we are using the 8th edition of the textbook, so the page numbers and section numbers do not match those at that website.
  • Recall that our test dates are: Sept 20, Nov 1, Dec 11, and the due dates of the Canvas quizzes will be announced in class and/or in the homework.
  • LAST REVISION: 11/30/23.

 

Aug 21 Check your Canvas notifications to check you can receive Canvas announcements.
Attendance will be noted, starting today.
Read course syllabus carefully. Make a note of the test dates in your calendar.
Review course website and repeat frequently during the semester.
Review the course’s Canvas portal.  

Read this study tip and read https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf for ideas on how to study most effectively.

Read pages xv-xvi and Section 0, and do Section 0 pgs 8-9: 1-3, 11, 29, 30, 41.
Read your lecture notes (meaning the notes you should have taken during lecture) and Section 1, and do 
Sec 1 pgs 17-19: 23, 27.
Aug 23 Read your lecture notes and Sections 1 & 2, and do
Sec 1 pgs 17-19: 34, 7, 42, and 
H1. Use Table 1.31 in the book on pg 17 to show that  b∗e = c  and  e∗b = b, and 
Sec 1 pgs 17-19: 1, 4, 5,  and
Sec 2 pgs 28-31: 16-18, 28, 32.
Aug 28 Read your lecture notes and Section 2 and do
Sec 2 pgs 28-31: 31, 37, 29, 10  and
H2. Does the determinant function
      d : ( { n × n  matrices with real entries }, matrix multiplication ) → ( ℝ , multiplication ),
      that sends a matrix to its determinant, satisfy the homomorphism property?  Is  d  an isomorphism?
H3. Does the natural logarithm function  ln : ( ℝ+ ,  multiplication ) → ( ℝ , + ), that sends a positive real     
       number to its natural logarithm, satisfy the homomorphism property?   Is  ln  an isomorphism?
H4. Let (G, ∗) denote an abelian group and let  f  denote the function  f(g) = g′ = the inverse of g,  for all g ∊ G.
       Does f satisfy the homomorphism property?  Is  f  an isomorphism?   What if G were not abelian?
Aug 30 Read your lecture notes and Section 3 and do Sec 3 pgs 38-39: 5, 7, 8, 11, 16, 17, 43.
Sep 04 Labor Day Holiday (no lecture)
Sep 06 Read your lecture notes and Section 4, and do
Sec 4 pgs 50-52: 3, 5, 9-12, 13(a)(b), 15, 16, 18, 24, 25, 33.
See the information sheet in Canvas for Canvas Quiz 1.
Canvas Quiz 1 opens today; it is due Sept 17.
Sep 11 Read your lecture notes and Section 5, and do
Sec 5 pgs 57-60: 9, 11, 15, 17, and
H5.  (a) Prove that the set   { 2n + 7m :  n, m ∊ ℤ }   is a subgroup of ℤ.
        (b) Prove that the set   { a n + b m :  n, m ∊ ℤ }   is a subgroup of ℤ for any (fixed) pair of integers a, b ∊ ℤ.
and 
Sec 5 pgs 57-60: 23, 27, 30, 35, 64.
Canvas Quiz 1 is due Sept 17.
Sep 13 Read your lecture notes and Section 6 and this file, and do
Sec 6 pgs 68-70: 42, 59, 3, 5, 45.
Canvas Quiz 1 is due Sept 17.
Test 1 will be in 1 week; see the information sheet in Canvas.
Sep 18 Read your lecture notes, Section 6, this file, and pages 70-71 and do
Sec 6 pgs 68-70: 24(a)(b), 56, 60, 17, 33, 36, and 
Sec 7 pg 74: 1, 3, 4, 7.
[ Optional:  read pages 72-74 and do pages 74-75: 9, 12, 16, 20. ]
Test 1 will be on Wednesday; see the information sheet in Canvas. 
Sep 20 Test 1 today; see the information sheet in Canvas.
Sep 25 Read your lecture notes and Section 8 and do Sec 8 pgs 84-87: 1-5, 8, 9, 13, 16, 35-37, 40.
Look over the solutions to Test 1 on Canvas; use password “math3321” to open the file.
Read through the suggestions of study techniques from Aug 21 above and see which one(s) might work for you.
Sep 27 Read your lecture notes and Section 8 and do Sec 8 pgs 84-87: 23-25, 17, 19, 51.
Oct 02 Read your lecture notes and Section 9 and do
Sec 9 pgs 94-97: 1, 5, 46, 15, 16  and also do 
Sec 10 pg 104: 39 (does not need material from Section 10).
Oct 04 Read your lecture notes and Section 9 and pages 97-99  and do
Sec 9 pgs 94-97: 23, 24, 25, 44   and also do 
Sec 10 pgs 102-104: 1, 6, 7.
Oct 09 Read your lecture notes and pages 97-99 and do 
Sec 10 pgs 102-104: 43, 33, H6, 34, 32, H7, H8    where H6-H8 are
H6. Suppose G is a group and H is a subgroup of G. If h ∊ H, prove that  hH = H = Hh.
H7. Suppose G is a group and H is a subgroup of G. Let a, b ∊ G. Prove that if aH = Hb, then aH = Ha and bH = Hb.
H8. Suppose G is a group and H is a subgroup of G. Let a ∊ G.
       (a)  Use the Subgroup Criterion to prove that  aHa-1  is a subgroup of G, where  aHa-1 = { aha-1  : h ∊ H}.
       (b)  Prove that   | aHa-1 |  = | H |.   
Oct 11 Read your lecture notes and pages 97-99 and do 
Sec 12 pg 120: 31(a)-(c)  (does not need material from Sections 11 or 12), and do 
Sec 10 pgs 102-104: 36, 42, 11-15, 41.
Watch this fun  TED talk  on symmetry and algebra (approx. 18 minutes).
Oct 16 Read your lecture notes and Section 12 and do   Sec 12 pgs 119-121: 1, 3, 8, 9, 15, 32.
Oct 18 Read your lecture notes and Section 12 and do
Sec 12 pgs 119-121: 24, 31(d), 42, H9, H10, 33, 36 (hint: see H8(b)), 39 (see Definition 12.19),   where H9 & H10 are  
  H9. For G = ( GL(n, ℝ),  matrix multiplication )  and   H = { a ∊ G : det(a) = 2m  ,  m ∊ ℤ },  prove  H ⊴ G.
H10. Suppose G is a group and let   H = { h ∊ G : ha = ah   for all  a ∊ G };   prove  H ⊴ G.
Remark: the subgroup H in question H10 is called the center of the group G.

See the information sheet in Canvas for Canvas Quiz 2.
Canvas Quiz 2 opens today; it is due Oct 29.
Oct 23 Read your lecture notes and do   Sec 13 pgs 129-131: 1, 3, 7, 6, 30, 37, 38.
Canvas Quiz 2 is due Oct 29.
Oct 25 Read your lecture notes and do   Sec 13 pgs 129-131:  36, 40, 15, 44.
Canvas Quiz 2 is due Oct 29.
Test 2 will be in 1 week; see the information sheet in Canvas.
Oct 30 Read your lecture notes and do
H11. Let G be a group. Justify that the map   (a, x) → xa-1 , for all x, a ∊ G, makes G into a G-set.
H12. Let G = S4 and let  X = { subsets of {1, 2, 3, 4} of cardinality 2 }.   For   x = {x1, x2} ∊ X & σ ∊ S4, define  σ(x) = { σ(x1), σ(x2) }.
        (a) Write out the 6 elements of X.
        (b) Describe explicitly how   (1   2)   and   (1   2   3)   act on the 6 elements of X.
        (c) Justify that   |σ(x)|=2   for all   σ ∊ S4   and for all   x ∊ X,   so that   σ(x) ∊ X   for all σ ∊ S4 and for all x ∊ X.
        (d) Justify that X is a G-set.

Test 2 will be on Wednesday; see the information sheet in Canvas. 
Nov 01 Test 2 today; see the information sheet in Canvas. 
Nov 06 Read your lecture notes and do   Sec 14 pgs 138-140:  20, H13, 6, 15, 21,    where H13 is
H13. Let G be a group, H ≤ G and   X = { xH : x ∊ G }.   Define    a(xH) = axH    for all a ∊ G, xH ∊ X (as discussed in lecture).
        (a) Justify that X is a G-set.
        (b) Justify that G acts transitively on X.

Look over the solutions to Test 2 on Canvas; use password “math3321” to open the file.
Read through the suggestions of study techniques from Aug 21 above and see which one(s) might work for you.
Nov 08 Read your lecture notes and do    Sec 14 pgs 138-140:  4, 29, 30.      
Nov 13 Read your lecture notes and Section 16 and do    Sec 16 pgs 148-149: 1, 2, 7, 9, 10.
Nov 15 Read your lecture notes and do    
Sec 17 pgs 155-6:  1, 2, 5, H14, 3, 4, 16, 19-21, 25, H15,   where  H14-H15 are:

H14. Justify that every group of order 32 is not simple.
H15. Justify that every group of order 50 is not simple.
Nov 20

Read your lecture notes. We did not quite manage to finish the lecture notes today, but the following result finishes the lecture notes.
THEOREM
Let p denote a prime number. Every finite p-group of order ≥ 2 is solvable.
Proof
Let G denote a p-group, where 2 ≤ |G| < ∞.   By results we proved from Section 17 of the book, we have that  |G| = pn  for some  n ∊ ℕ. 
By (a) of the 1st Sylow Theorem, there exists   Hi ≤ G of order  pi  for all i ∊  {1, …, n}.  By (b) of the 1st Sylow Theorem, we can choose the Hi such that  Hi ◃  Hi+1  for all i ∊  {1, …, n−1}.  In particular, the factor groups  Hi+1 ⁄ Hi   exist for all i ∊  {1, …, n−1}.   Since 
| Hi+1 ⁄ Hi |  =  | Hi+1 |  ⁄  | Hi | = pi+1 ⁄ pi  =  p,  it follows that  Hi+1 ⁄ Hi  ≅  ℤp  for all i ∊  {1, …, n−1}, which implies that  Hi+1 ⁄ Hi   is abelian and simple (since p is a prime number) for all i ∊  {1, …, n−1}. Hence, G is solvable.   ◼

Do 
H16.  (a) Justify that every group of order 12 is not simple.
          (b) Justify that every group of order 56 is not simple.
H17.  (a) Justify that every group of order 35 is cyclic.
          (b) Justify that every group of order 159 is cyclic.
H18. (a) Find a composition series for ℤ48 & determine the list of factor groups for your series (see pg 164 #7).
         (b) Find a second composition series for ℤ48 & determine the list of factor groups for your series (see pg 164 #7).
         (c) Compare your lists of factor groups in (a) & (b) (you should find they are the same groups & occur the same number of times).
Also, do   Sec 18 page 164:  19.

On Monday, the plan is to teach an application of group theory to tasks that we perform most days. 

Please remember to complete online the student feedback survey by 11 pm on Dec 5 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the information sheets and/or the solution sheets for the tests, the examples provided in class, the Canvas graded homework (CQ1, CQ2 & CQ3), corniness of jokes …??). Thank you!
Nov 22 No lectures nor office hours today; UTA offices open until noon.
See the academic schedule at     https://www.uta.edu/academics/academic-calendar
Nov 27 Read your lecture notes from today’s lecture about applications of group theory & reflect on the uses of group theory!!

See the information sheet in Canvas for Canvas Quiz 3.
Canvas Quiz 3 opens today; it is due Dec 6.

Please remember to complete online the student feedback survey by 11 pm on Dec 5 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the information sheets and/or the solution sheets for the tests, the examples provided in class, the Canvas graded homework (CQ1, CQ2 & CQ3), corniness of jokes …??). Thank you!
Nov 29 We will spend today’s lecture reviewing the course material & addressing students’ questions; no new homework.

Canvas Quiz 3 is due Dec 6.

Please remember to complete online the student feedback survey by 11 pm on Dec 5 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the information sheets and/or the solution sheets for the tests, the examples provided in class, the Canvas graded homework (CQ1, CQ2 & CQ3), corniness of jokes …??). Thank you!

I will have my usual office hours through Dec 4 inclusive, and I will also have additional office hours as noted below.
Dec 04 We will spend today’s lecture reviewing the course material & addressing students’ questions; no new homework.

Canvas Quiz 3 is due Dec 6.

Please remember to complete online the student feedback survey by 11 pm on Dec 5 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the information sheets and/or the solution sheets for the tests, the examples provided in class, the Canvas graded homework (CQ1, CQ2 & CQ3), corniness of jokes …??). Thank you!

The Final Test will be on Monday Dec 11; see Canvas for an information sheet.

I will have my usual office hours through today inclusive, and I will also have additional office hours as noted below.

Dec 06
NOON

Office hour today noon-12:55 PM online only – see Canvas announcements for the Teams link.
This is time the instructor is planning to be on Teams for students to “drop by” to ask questions.

Dec 08
1:20 PM

Office hour today 1:20-2:20 PM online only – see Canvas announcements for the Teams link.
This is time the instructor is planning to be on Teams for students to “drop by” to ask questions.

Dec 09
1:20 PM

Office hour today 1:20-2:20 PM online only – see Canvas announcements for the Teams link.
This is time the instructor is planning to be on Teams for students to “drop by” to ask questions.

Dec 11
5:30 PM

FINAL TEST today, starting at 5:30 PM; see Canvas for an information sheet.
Look over Tests 1-2 and Canvas Quizzes 1-3 and their solutions posted on Canvas to study for this test.
(Pay attention to how the solutions on the solution sheets are written; e.g., the level of detail provided in each solution.)

Recall that the assignments from Fall 2019 can be viewed here .