Homework for MATH 3321 in Spring 2025


return to Spring 2025 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 from Fall 2023 can be viewed here; in particular, it will give you a rough outline of what material will be presented and how many lectures will be spent on each item.
  • Recall that our test dates are: Feb 13, Mar 27 and May 1, and the due dates of the Canvas quizzes will be announced in class and/or in the homework.
  • LAST REVISION: 4/08/25.
Jan 14 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.
Jan 16 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 do
Sec 1 pgs 17-19: 1, 4, 5   and  Sec 2 pgs 28-31: 16-18, 28, 32.
Jan 21 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?
Jan 23 Read your lecture notes and Section 3 and do  Sec 3 pgs 38-39: 5, 7, 8, 11, 16, 17, 43. 
See the information sheet in Canvas for Canvas Quiz 1.
Canvas Quiz 1 opens on Jan 29; it is due Feb 9.
Jan 28 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 tomorrow; it is due Feb 9.
Jan 30 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.

By this time, you have seen quite a few theorems. It is best NOT to memorize the theorems, but to do enough of the homework so that the results of the theorems become known to you, though perhaps with different wording or using pictures.

Canvas Quiz 1 is due Feb 9.
Feb 04 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 Feb 9.
Feb 06 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. ]
Canvas Quiz 1 is due Feb 9.
Test 1 will be in 1 week; see the information sheet in Canvas.
Feb 11 Read your lecture notes and Section 8 and do  Sec 8 pgs 84-87: 1-5, 8, 9, 13, 16, 35-37, 40.
Test 1 will be on Thursday; see the information sheet in Canvas. 
Feb 13 Test 1 today; see the information sheet in Canvas.
Feb 18 Read your lecture notes and Section 8 and do Sec 8 pgs 84-87: 23-25, 17, 19, 51.
Look over the solutions to Test 1 on Canvas; use password “math3321” to open the file.
Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
Feb 20 Read your lecture notes and Section 9 and do  Sec 9 pgs 94-97: 1, 5, 46, 15, 16, 13, 39
and also do  Sec 10 pg 104: 39 (does not need material from Section 10).
Feb 25 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.
Watch this fun  TED talk  on symmetry and algebra (approx. 18 minutes).
Feb 27 Read your lecture notes and pages 97-99 and do 
Sec 10 pgs 102-104: 43, 33, H6, 34, 32, H7, H8,
Sec 12 pg 120: 31(a)-(c)  (does not need material from Sections 11 or 12),  and 
Sec 10 pgs 102-104: 36, 42, 11-15, 41,       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 |.
Mar 04 Read your lecture notes and Section 12 and do   Sec 12 pgs 119-121: 1, 3, 8, 9, 15, 32.
Mar 06 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 on Sat March 15; it is due Monday March 24.
Mar 11
&
Mar 13		 SPRING BREAK
Go over Quiz 1, the worksheet (from before Test 1) and Test 1 and their solutions on Canvas & read over all lecture notes & get caught up on watching any lecture recordings/videos and the above TED talk (see Feb 25 above). Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you. Get caught up on all homework!!
Mar 18 Read your lecture notes and do   Sec 13 pgs 129-131: 1, 3, 7, 6, 30, 37, 38.
Canvas Quiz 2 is due March 24.
Mar 20 Read your lecture notes and do   Sec 13 pgs 129-131:  36 (hint: see Sec 10 #41 assigned earlier), 40, 15, 44.
Canvas Quiz 2 is due March 24.
Test 2 will be in 1 week; see the information sheet in Canvas.
Mar 25 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 Thursday; see the information sheet in Canvas. 
Mar 27 Test 2 today; see the information sheet in Canvas. 
Apr 01 Read your lecture notes and do
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.
and also do Sec 14 pgs 138-140:  20.
Look over the solutions to Test 2 on Canvas; use password “math3321” to open the file.
Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
Apr 03 Read your lecture notes and do    Sec 14 pgs 138-140:  6, 15, 21, 4, 29, 30.     
Apr 08 Read your lecture notes and Section 16 and do    Sec 16 pgs 148-149: 1, 2, 7, 9, 10.
Apr 10 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.
Apr 15 Read your lecture notes and 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.
Apr 17 Read your lecture notes and do
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.
See the information sheet in Canvas for Canvas Quiz 3.
Canvas Quiz 3 opens today; it is due Saturday April 26.
Apr 22 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; it is due Saturday April 26.
Please remember to complete online the student feedback survey by 11 pm on April 29 — 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), jokes sufficiently corny??….??).  Thank you!
Apr 24 We will spend today’s lecture reviewing the course material & addressing students’ questions; no new homework.
See the information sheet in Canvas for Canvas Quiz 3; it is due Saturday April 26.

Please remember to complete online the student feedback survey by 11 pm on April 29 — 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), jokes sufficiently corny??….??).  Thank you!
Apr 29 We will spend today’s lecture reviewing the course material & addressing students’ questions; no new homework.

Please remember to complete online the student feedback survey by 11 pm today — 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), jokes sufficiently corny??….??).  Thank you!

The Final Test will be on Thursday May 1; see Canvas for an information sheet.
Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.

I will have my usual office hours through today inclusive; after today, I will have office hours as noted below.
Apr 30 Office hour today 4 pm – 6 pm in PKH 462.  This is time the instructor is planning to be in PKH 462 for students to drop by to ask questions.
May 01 Office hour today 2 pm – 4 pm in PKH 462.  This is time the instructor is planning to be in PKH 462 for students to drop by to ask questions.
May 05 Office hour today 1 pm – 2 pm online only – see the information sheet for the Final Test for the Teams link.
This is time the instructor is planning to be on Teams for students to “drop by” to ask questions.
May 1
5:30 PM		 FINAL TEST today, starting at 5:30 PM; see Canvas for an information sheet.
Look over Tests 1 & 2, the worksheet just prior to Test 1 and Canvas Quizzes 1-3 and their solutions posted on Canvas when you 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.)
Read through the suggestions of study techniques from Jan 14 above and see which one(s) might work for you.
   
   
  The rows below are from Fall 2023 and will be edited and moved above this row as the semester progresses.
The homework from Fall 2023 can be viewed in its entirety here.
   
   
 

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.   ◼
   
   

The homework from Fall 2023 can be viewed in its entirety here.