Math 301 Abstract Algebra

New Information

Final exam Monday May 1, 9:45-11:45 AM in 004 Carver (regular classroom).

Old Final


Index

Homework assignments

Homework will be assigned during each class period and discussed during the next class period, so you should attempt all the problems before then.   All material assigned Friday-Wednesday will be collected the following Monday.  Some problems (marked with * below) will be graded; there will also be a completeness grade.  The first homework collection (covering problems assigned Mon. Jan. 9 - Wed. Jan 18) will be Monday Jan. 23.
 
Date  Chapter Problems
(* to be graded)		 Section(s) covered
M J 9
 0
 1 (5, 12, 25), 2, 3, 4, 7, 8*, 12, 16*
 Properties of Integers
W J 11
 0
 9, 46, 47, HH1*
 Equivalence relations and modular arithmetic (including additional material not in  Ch. 0)
F J 13
 0
18, 19, HH2
 Functions (see also Function Theorem below)
W J 18
 1
 1,2,3
 symmetries of a square
F J 20
2
 1, 5*, 6*, 7, 9, 11
 all
M J 23
 3
 1(Z6, U(12), D4), 4*, 5, 6 and
Ch 2:17, 24, 26*		 terminology, use of subgroup tests, some examples
W J25
 3
 8, 10, 11ab, 16*, 17b, 21, 22, 28
 all now covered
F J27
 4
 1, 2*, 5, 7, 11, 13, 14
 up to classification
M J29
 4
 16*, 17, 18, 19, 30, 31, 42
 all now covered
W F1
 5
 17
 introductory material
F F3
 5
 18a, HH3
M F6
 5
 1,3,4,10,19,20,21,22
 through p. 103
M F13
 6
 3, 4, 5, 7, 22*, 25
 motivation, defs and examples, properties
W F15
 6
 10, 11, 12*, 20, 27, 29, HH4*
 properties, Cayley's Thm, automorphisms
F F17
 6
 26, 32-35
 all of chapter 6
M F20
 7
 1-5
W F22
 7
 7,8,10*,14,15,17,18,19,HH5*,HH6*
 up to classification of order 2p
F F24
 7
 HH7, HH8*
 D_n by generators and relations
M F27
 8
 4*, 5, 6, 8; Ch 7: 20, 21, 24, 26*, 27
W M1
 8
 7, 10*, 11, 116, 19
 through mid p. 154
F M3
 11
 10, 15, HH9, extra credit XC1
 classification of finite abelian groups
M M20
 9
 1, 4*, 7, 8a, 11, 27
 normal subgroups, internal direct products
W M22
 9
 28, 29, 30*, 31, 32, 44c* (first list the distinct left cosets and their elements, then make Cayley table), HH10*
 factor groups
F M24
 9
 10*, 13, 16, 18, 19, 37, 46*
 all
M M27
 10
def & ex of homs
W M29
 10
 2, 7, 8*, 9, 13, 14
 properties of homs
F M31
 10
15, 16, 26, 32, 43, 48
 all
W A5
 12
 1,2*,4,11, HH11
 def and ex
F A7
 12
 5, 6*, 7, 12, 13, HH12*
 all
F A14
 14
 4*,7,9,12,14,15, 16*, 17, HH13
 ideals
M A17
 14
 19, 23, 24, 35, 38*, 39, 47, HH14*
 factor rings
W A 19
 14
 27, 30, 31, 32, HH15*
 all
HH1: Prove multiplication [a][b]=[ab] is well-defined on Z_n, i.e. prove if [a]=[a'] and [b]=[b'] then [ab]=[a'b'].
HH2: Prove part (3) of the Function Theorem.
HH3: f=(123), g=(12) h=(14).  Compute: fg, hfg, fgh, f^-1
HH4: Let G be isomorphic to H.  Prove Z(G) is isomorphic to Z(H).
HH5: List all the distinct left cosets of H={e, (123), (132)} in A_4.  For each coset list the elements.
HH6: Let G be a group, H < G, a,b in G. Prove that Ha = Hb iff ab^-1 in H.
HH7: list orders of all elements of D_6
HH8: In D_5: a) |a^2 b|=? b) a^2 b a b=? (as a^k b) 
         c) List all the distinct left cosets of H={e, ab} in D_5.  For each coset list the elements.
HH9: a) Give a list of nonisomorphic abelian groups of order  360 such that any abelian group of order 360 is isomorphic to a group on you list.  b) To what group on the list is Z_20 dirsum Z_18 isomorphic?
XC1: Theorem 8.3 states that if GCD(s,t)=1 then U(st) isomorphic U(s) dirsum U(t).  Is the converse true?  i.e., is it true that if U(st) isomorphic U(s) dirsum U(t), then GCD(s,t)=1?
HH10: Let G be the internal direct sum of subgroups J and K (i.e., G = H +int K)
Define f: H +ext K -> G by f(h,k)=hk.  Prove f is 1-1.
HH11: Make a conjecture about for which n Z_n has nonzero zero divisors (i.e. nonzero elements a,b such that ab=0).  Support your conjecture with examples from Z_3, Z_4, and perhaps Z_5, Z_6.
HH12: Complete the prrof that L is a ring by proving the distributive properties (#6 in the defintion of a ring).
L=R^2 with ordinary (vector space) addition and multiplication defined by
[a] [c] = [a(c+d)]
[b] [d] = [b(c+d)]
HH13: Let R, S be rings.  R dirsum S = {(r,s) : r in R, s in S} with comordinatewise operations. 
R'={(r,0) : r in R} is a subset of R dursum S.  Show R' is an ideal of R dirsum S.
HH14: In this question R means real #s.  Let K=<x^2-3x+2> ideal of R[x].  In R[x]/K compute
(all answers must be in standard form i.e., ax+b + K where a, b are real #s):
a) 2x^4+2x+2 + K (simplify to standard form)
b) i) (x+1 + K) + (3x-5 + K)
    ii) (x+2 + K) + (x-1 + K)
c) i) (x+1 + K) (3x-5 + K)
    ii) (x+2 + K) (x-1 + K)
HH15: Prove that in a commutative ring with unity, a maximal ideal must be a prime ideal.

Course Information

Course Objectives Math 301 provides an introduction to abstract algebraic structures, including a thorough study of groups and an introduction to rings. In this course students will develop the ability to work with abstract structures and will expand their ability to write proofs.  More detailed objectives for each examination will be available shortly before the exam.

Syllabus

Policies

Meeting Time and Place

Lectures
MWF 10:00 PM in Carver 4

Text

Joseph A. Gallian, Contemporary Abstract Algebra, 5th Ed.
Chapters 0 - 10 and 12 will be covered.  Selected material from Ch 11, 13 and 14 will be covered. 

Instructor Information

Examination Information

Final exam Monday May 1 9:45-11:45 AM in 004 Carver (regular classroom).

Old Final

Solutions to Test 3

Old Test 3
Test 3 will cover Chapters 9 and 10 (note there is a lot of related material in 6, 7, 8).

Review in class Wed. March 8 and test study session Thursday March 9 1:30-2:30 in Carver 18.
Test Friday March 10 will cover Ch 6-8 and a bit of 11.
Old Test 2
Old Test 2 Solutions

Review in class Wed. Feb. 8 and test study session Thursday Feb. 9 1:30-2:30 in Carver 298.
First exam  Friday, Feb. 10 and will cover Ch 0 - Ch 5.
The best preparation for the test is working problems.  The first choice of problems to work is all the assigned homework problems in the text (not just the graded ones) and problems mentioned as homework in the lecture (fill in parts not done). Note the old test below does NOT cover all the material we have covered.

Solutions to Test 1
Solutions to old test 1
Extra review problems (for test 1)

Selected Mathematical Content from Class

Function Theorem.  Let A, B, C, D be non-empty sets.  Let f: A-> B, g: B -> C, h: C -> D be functions.
1. (hg)f = h(gf)
2. if f and g are 1-1 then gf is 1-1
3. if f and g are onto then gf is onto    (H2: prove this)
4 a) If f is 1-1 then there exists function k: B -> A such that kf = i_A
   b) If there exists function k: B -> A such that kf = i_A then f is 1-1
   c) If f is onto then there exists function j: B -> A such that fj = i_B
   d) If there exists function j: B -> A such that fj = i_B then f is onto
   e) If kf = i_A and fj = i_B then k = j (called f^-1)

Mathematical notation and proof writing


Iowa State University complies with the American with Disabilities Act and Section 504 of the Rehabilitation Act.  If a student has a disability that qualifies and requires accommodations, he/she should contact the Disability Resources (DR) office for information on appropriate policies and procedures. DR is located on the main floor of the Student Services Building, Room 1076; their phone is 515-294-6624. Any student who requires an accommodation under such provisions should contact the instructor privately as soon as possible and no later than the end of the first week of class or as soon as documentation of the need for accommodation is obtained. Contact may be made by e-mail (LHogben@iastate.edu), telephone (4-8168), or in person (office 488 Carver).  No retroactive accommodations will be provided in this class.
 
 
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