MWF 11:3012:25 BO128, TR 11:3012:25 HA012
Instructor: Sasha Leibman  office: MW406 office hours: Wednesday 23pm, Thursday 12pm 
email: leibman.1@math.osu.edu  
phone: 6142920663 
Textbook: D.S.Summit and R.M.Foote, Abstract Algebra, 3rd edition
Midterm 1 — September 28.
Solutions.
List of topics.
Sample problems.
Midterm 2 — November 1.
Solutions.
List of topics.
Sample problems.
There will be no Midterm 3, sorry.
The final exam scheduled days are December 11 and December 14, of your choice.
Homework:
HW1  recommended:  by Friday, August 25   Exercises  1.1.1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 27, 28, 29, 32, 33, 36 (pages 2123). 
written part:  due by Tuesday, August 29   Exercises  1.1.9, 22, 25, 30, 31 (pages 2223).  
HW2  recommended:  by Friday, September 1   Exercises 
1.2.3, 4, 5, 6, 8, 9–13, 14, 15 (page 28);
1.3.1, 4, 5, 6, 7, 8, 9, 13, 16, 18 (pages 3234); 2.1.3, 4, 7, 8, 9, 11, 12, 14, 15 (pages 4849); 2.3.1, 2, 3, 13, 16 (pages 6061); 2.4.1, 2, 8, 9, 10, 11, 13, 14, 15, 19, 20 (pages 6566). 
written part:  due by Tuesday, September 5   Exercises  1.2.7 (page 28); 1.3.15 (page 33); 1.5.3 (page 36); 2.1.6 (page 48); 2.3.12 (page 60); 2.4.6, 7 (page 65).  
HW3  recommended:  by Friday, September 8   Exercises 
2.3.25 (page 61);
3.1.3, 4, 5, 15, 16, 17, 22, 23, 24, 25, 27, 28, 29, 30, 31 (pages 8589); 3.2.1, 2, 3, 6, 11, 12, 18, 20, 23 (pages 9596). 3.5.2, 3, 4, 7 (page 111). 
written part:  due by Tuesday, September 12   Exercises  3.1.14(a,b,c,d), 34(a,b), 42 (pages 8889); 3.2.5(a,b), 8, 10 (pages 9596), 3.5.5 (page 111).  
HW4  recommended:  by Friday, September 15   Exercises 
1.6.1, 2, 4, 9, 11, 15, 16, 17, 19, 22 (pages 3941);
3.1.6, 7, 8, 10, 12, 32, 37 (pages 8586); 3.3.7, 8, 10 (page 101). 
written part:  due by Tuesday, September 19   Exercises  1.6.18 (page 40); 3.1.9, 33, 36, 38 (pages 8588); 3.3.3, 4 (page 101).  
HW5  recommended:  by Friday, September 22   Exercises 
1.7.1, 2, 3, 8, 9, 10, 12, 14, 15 (pages 4445);
4.1.4, 5, 6, 10 (pages 116117); 4.2.1, 3, 7, 9, 10, 11 (page 121122). 4.3.2, 6, 7, 8, 9, 11, 13, 19, 20, 21, 22, 24, 25, 27, 30, 31, 32, 33, 36 (page 130132). 
written part:  due by Tuesday, September 26   Exercises  1.7.23 (page 45); 4.1.3, 9(a,b) (page 117); 4.2.4, 6, 8 (pages 121122); 4.3.34 (page 132).  
HW6  recommended:  by Friday, October 6   Exercises 
3.4.1, 7, 9, 10 (page 106);
5.1.1, 2, 3, 7, 15, 17, 18 (pages 156158); 5.2.1, 2, 3, 4, 5, 7, 9, 10, 13, 14 (page 165167). 
written part:  due by Tuesday, October 10   Exercises 
3.4.2 (the Q_{8} part only), 5 (page 106);
5.1.14 (page 157);
5.2.1(a,b), 4(b) (pages 165166);
and two more problems: (1) Prove that any pgroup is solvable (where p is a prime); (2) If groups A and B have relatively prime orders, prove that any subgroup of A×B has the form H×K where H≤A and K≤B. 

HW7  recommended:  by Monday, October 16   Exercises  4.4.1, 6, 7, 12, 17, 1819 (pages 137139). 
written part:  due by Tuesday, October 17   Exercises  4.4.5, 8(a,b,c), 13 (pages 137138).  
HW8  recommended:  by Friday, October 20   Exercises 
4.5.1, 2, 4, 6, 9, 13, 16, 18, 19, 24, 26, 29, 30, 32, 33, 34, 36 (pages 146148);
5.5.1, 2, 4, 5, 14, 18 (pages 184186). 
written part:  due by Tuesday, October 24   Exercises  4.5.7, 17, 35, 37(a,b,c,d) (pages 146148); 5.5.6, 9, 13 (pages 184186).  
HW9  recommended:  by Tuesday, November 7   Exercises 
7.1.122 (pages 230232);
7.2.19, 12 (pages 237239). 
written part:  due by Wednesday, November 8   Exercises  7.1.14(ad), 15, 21(a,b) (pages 231232); 7.2.2, 10(a,c) (pages 238239).  
HW10  recommended:  by Tuesday, November 14   Exercises 
7.3.116, 34, 35 (pages 247250);
7.4.113, 14, 16, 18, 19, 23, 24, 26, 27 (pages 256259). 
written part:  due by Wednesday, November 15   Exercises  7.3.17, 20, 22(a,b), 33(a,b) (pages 248250); 7.4.15(a,b,c), 17, 33(a,b,c) (pages 257259).  
HW11  recommended:  by Monday, November 20   Exercises  8.2.1, 2, 3, 6, 7, 8 (pages 282283). 
written part:  due by Tuesday, November 21   Exercises  8.2.5(a,b,c) (page 283); 8.3.8(a,b,c) (page 293).  
No HW for Thanksgiving  
My solutions to HW1, HW2, HW3, HW4, HW5, HW6, HW7, HW8, HW9, HW10, HW11
Studied topics:
August 22:  Definition and examples of groups (Sections 1.1 and 0.3) 
August 23:  Examples of groups (Sections 1.1, 1.2) 
August 24: 
Elementary properties of groups (Section 1.1)
Order of elements in a group (Section 1.1) Direct products of groups (Example 6 on p.18) 
August 25:  Exercises from Section 1.1 
August 28: 
Classification of groups of orders 1–6.
Subgroups. Subgroups generated by subsets (Sections 2.1, 2.4). 
August 29: 
Cyclic groups and their subgroups (Section 2.3)
Groups defined by generators and relations (Section 1.2) 
August 30: 
The dihedral group D_{2n} (Section 1.2)
The quaternion group Q_{8} (Section 1.5) Finite matrix groups (Section 1.4) 
August 31:  The symmetric group S_{n}. Parity of permutations (Sections 1.3 and 3.5) 
September 1:  Exercises from Sections 1.2, 2.4. 
September 5: 
Subgroups and generators of finite cyclic groups.
The groups Z_{n}^{*} (Section 0.3)
The alternating group A_{n} (Section 3.5) 
September 6: 
Cosets. Index of a subgroup. (Sections 3.1, 3.2)
Lagrange's theorem and corollaries (Section 3.1) Counting principles (Section 3.2) 
September 7: 
Normal subgroups (Section 3.1)
Factorization, quotient groups (Section 3.1) 
September 8:  Groups, defined by generators and relations  revisited. 
September 11: 
Conjugation (Section 4.3)
Properties of normal subgroups (Section 3.1) The normalizer and the centralizer of a subgroup (Sections 2.2 and 3.1) Homomorphisms of groups (Section 1.6) 
September 12: 
Properties of homomorphisms (3.1)
The kernel of a homomorphism. Isomorphisms (Section 3.1) 
September 13: 
The 1st isomorphism theorem (Section 3.3)
Representation of any group as a factor of a free group. Reduction of a homomorphism to a quotient group (page 100). 
September 14: 
The 2nd and the 3rd isomorphism theorems (Section 3.3)
The lattice of subgroups of a group. The 4th isomorphism theorem (Section 3.3) 
September 15:  Exercises from Sections 1.6, 3.1, 3.3. 
September 18:  Group actions (Sections 1.7, 4.1) 
September 19: 
Orbits and stabilizers of a group action (Sections 1.7, 4.1)
Actions of a group on itself – by left multiplication (Section 4.2) Cayley's theorem (Section 4.2) 
September 20: 
Right group actions (Section 4.3)
Actions of a group on itself – by conjugations. Conjugacy classes. Normalizers and centralizers (Section 4.3) 
September 21: 
Conjugacy classes in S_{n} (Section 4.3)
Simple groups (Section 4.3) 
September 22:  Exercises from Sections 1.7, 4.1, 4.2, 4.3. 
September 25: 
Exercises from Section 4.3.
The simplicity of A_{n} for n≥5 (Section 4.3) 
September 26:  Review. 
September 27:  Review. 
September 28:  Midterm 1. 
September 29:  Subnormal and composition series. JordanHölder theorem. Hölder's program (Section 3.4) 
October 2: 
Solvable groups (Section 3.4)
Direct products of groups. Internal direct products of subgroups (Sections 5.1, 5.4) 
October 3: 
Internal direct products of subgroups (Sections 5.1, 5.4)
The Chinese remainder theorem (Proposition 5.6 on p.163) 
October 4: 
Direct products and sums of infinite collections of groups (Section 5.1)
The fundamental theorem of finite abelian groups (Section 5.2) 
October 5:  The fundamental theorem of finite abelian groups (Section 5.2) 
October 6:  Exercises from Section 5.2. 
October 9: 
Exercises from Sections 3.4, 5.1.
Groups of automorphisms of groups (Section 4.4) The groups of automorphisms of Q_{8}, Z_{n}, D_{2n} (Section 4.4) 
October 10:  Inner and outer automorphisms (Section 4.4) 
October 11: 
Groups of automorphisms of
S_{n}, Z_{p}^{n}
(Section 4.4)
Characteristic subgroups (Section 4.4) Semidirect products of groups (Section 5.5) 
October 16:  Internal and external semidirect products (Section 5.5) 
October 17: 
Examples of external semidirect products (Section 5.5)
Sylow's theorems (Section 4.5) 
October 18:  Sylow's theorems (Section 4.5) 
October 19: 
Examples of applications of Sylow's theorems (Section 4.5)
Groups of orders pq, pqr (Sections 5.3, 5.5) 
October 20:  Exercises from Section 4.5. 
October 23:  Exercises from Sections 4.5, 5.5 
October 24:  Some problems... 
October 25: 
The group of rotations of a cube.
Groups of orders pq^{k}, p^{2}q (Sections 5.3, 5.5) 
October 26: 
Commutator calculus, derived groups (Sections 5.4, 6.1)
The derived series and solvable groups (Section 6.1) 
October 27:  The upper and the lower central series. Nilpotent groups (Section 6.1) 
October 30: 
Finite nilpotent groups (Section 6.1)
Subgroups of a free group (Section 6.3) 
October 31:  Review. 
November 1:  Midterm 2. 
November 2: 
Rings — definition and examples (Sections 7.1, 7.2)
Elementary properties of rings (Section 7.1) Zero divisors and units in a ring (Section 7.1) 
November 3: 
Quaternions, Boolean rings, rings of endomorphisms (Section 7.2)
Finite rings without zero divisors (cf. Section 7.1) Integral domains and their characteristic (Section 7.1) Fields of fractions (Section 7.5) 
November 6: 
Universality of the field of fractions (Section 7.5)
Rings of fractions (Section 7.5) Quotient rings and ideals (Section 7.3) 
November 7: 
Examples of ideals.
Exercises from Section 7.1. 
November 8: 
Homomorphisms of rings (Section 7.3)
Isomorphisms of rings. Isomorphism theorems for rings (Section 7.3) 
November 9: 
Properties of ideals (Section 7.4)
Ideals and fields (Section 7.4) Ideals, generated by sets. Principal ideals (Section 7.4) 
November 13: 
"Divisibility" of ideals. gcd and lcm of two ideals (Section 7.4)
Maximal and prime ideals (Section 7.4) 
November 14: 
Properties of prime ideals (Section 7.4)
Existence of maximal ideals via Zorn's lemma (Section 7.4 and Appendix I) The nilradical is the intersection of prime ideals (Exercise 26 on p.258) 
November 15: 
Radical ideals (Exercises 30 and 31 on p.258)
Primary ideals (Exercise 41 on p.260) Noetherian rings (p.316) 
November 16: 
The ring of polynomials over a Noetherian ring is Noetherian (Hilbert's Basis Theorem, p.316)
The primary decomposition theorem (p.684) Prime and irreducible elements of an integral domain (Section 8.3) 
November 17: 
Rings of quadratic integers (Sections 7.1, p.229, and 8.3)
Unique factorization domains (Section 8.3) 
November 20:  Principal ideal domains (Section 8.2) and Unique factorization domains (Section 8.3) 
November 21:  Euclidean domains (Section 8.1) 
Plans for the nearest future:
The DedekindHasse norm (Section 8.2)
Primes in the ring of Gaussian integers Z[i] (Section 8.3) 