# MATH 5401 - Abstract Algebra I

Class Information:

Handouts:

• Some basic number theory facts

Student notes from Fall 2013 (thanks Sam!):

More Resources:

Study guides:

Test solutions:

Homework:

 1.1 # 5,8,11,12,25,28 - Sam's solutions - Extra solution to 28 1.2 1(b,c), 4   A) Compute the group table for D_6 and D_8.   B) Find the inverse of r^2 in D_6, r in D_8, r^2 in D_8, and sr in D_8.  What is the inverse of r in D_2n?  What is the inverse of sr^i in D_2n? C) In D_8, simplify the following products:  s r^2 s r^3 and r s r^(-2). - Sam's solutions 4550 HW Do these problems from MATH 4550: HW 1 - 1, 4, 5, 7, 12, 14 1.3 1, 2, 4(a)   - Sam's solutions 4550 HW Do these problems from MATH 4550: HW 3: 1, 2, 4, 5 1.6 1, 2, 3, 4, 5, 6, 13, 15   - Sam's solutions - Extra solutions to 1,3,4,5,6,15 are here.   Extra solutions to 2 and 13 are here. 1.7 14, 15, 16, 17, 18   - Sam's solutions - Extra solutions to 14, 15, 16 are here.   Extra solutions to 17 and 18 are here. 4550 HW Do these problems from MATH 4550: HW 5: 1, 2, 3 2.1 1(b), 2(a), 3(b), 10(a), 11, 12(b), 14   - Sam's solutions - Extra solution to 11 is here.  Extra solutions to the problems except for 11 are here. 2.2 2, 5(a), 6, 7, 11   - Sam's solutions - Other solutions are here. 2.3 11, 12, 13, 26 Note: For #26, in the book for this problem, Z_n denotes an abstract cyclic group of order n.  So the notation x^n does not mean multiplication in the group Z_n = {0,1, ..., n - 1}, it means multiplication in the abstract cyclic group.  Also, do:  A) Find all the generators for Z_8 where Z_8 = {0,1,2,3,4,5,6,7} is the integers modulo 8 under addition.   - Sam's solutions - Other solutions are here. 4550 HW Do these problems from MATH 4550: HW 2: 1, 2 (not U3), 4, 6, 11, 12, 13, 14HW 3: 3HW 4: 3, 5, 6, 8, 12, 13, 15, 16(c,d)HW 7: 1, 3, 4, 6 TEST 1 covers everything above 3.1 1, 3, 4, 5, 20, 22(a), 36A) Calculate the elements of ( Z_2 x Z_4 ) / < (0,1) >B) Calculate the elements of  D_8 / C) Give the order of the element (2,1) + <(1,1)> in the factor group (Z_3 x Z_6) / <(1,1)>.   - Sam's solutions - Extra solutions for the problems (except for 5 and 20) are here.   Extra answers to 5 and 20 are here. 3.2 4, 5, 8, 16   - Sam's solutions - Extra solutions for the problems (except for 5) are here.   Extra answer to 5 is here. 3.3 For A and B, find groups that the following are isomorphic to and use the first isomorphism theorem to prove it:A) ( Z_2 x Z_4 ) / < (0,1) >B) ( Z x Z ) / < (1,2) > C) Let G and G' be groups, and let H and H' be normal subgroups of G and G', respectively.  Let f be a homomorphism from G to G'.  Show that f induces a natural homomorphism                           g : (G / H) --> (G' / H') if f(H) is a subset of H'   - Sam's solutions - Extra solutions here 3.5 2   - Sam's solutions - Extra solutions here 4550 HW Do these problems from MATH 4550:   HW 6: 1, 2, 3, 4HW 8 : 1, 2, 3, 6, 7HW 9: 1--9 TEST 2 covers up to this point 4.3 2(a), 4, 8   - Sam's solutions - Extra solutions here (Note: there is a typo in the solution for #8 at the very end.  It should say sigma o tau /= tau o sigma.    Right now it has sigma o tau = tau o sigma.) 4.4 2, 15   - Sam's solutions - Extra solutions here 4.5 5, 13, 30 Hint for 5: Show that is the unique Sylow p-group of size p^k if p is odd and 2n = 2p^k * m (where p does not divide m).   - Sam's solutions - Extra solutions for 5 and 30 are here.   Extra answers to 13 are here 5.2 A) Let G be a finite abelian group.  Prove that G is simple if and only if G is isomophic to Z_p for some prime p. B) Find all abelian groups of size 36.  Find all abelian groups of size 540.    - Sam's solutions - Extra solution to A is here 5.4 A) Classify all groups of size 5^2 * 7    - Sam's solutions - Extra solution here

Lecture notes: (we didn't start taking pictures until 9/19)