Algebra

Office Hours:

Prerequisite:

Course description:

Courseload:

Bibliography:

• Dummit and Foote - Abstract Algebra. (Wiley; 3rd edition)

Course Notes:

• Updated as we go.. NB: These notes were written on little sleep, and are full of (sometimes horrific) mistakes, some of which we caught in class (but have not been corrected in the notes..) CAVEAT EMPTOR! Part 1, Part 2, Part 3, Part 4, Part 5

Homework Assignments:

Week 1:

• DF Chapters 1.3: 15,16. 1.4: 7,11. 1.6: 10,13,17,20 1.7:15,16,21
• Prove that disjoint cycles give commuting elements in the symmetric group S_n.
• Show that the decomposition of a permutation as a product of disjoint cycles is unique up to rearrangement.
• Show that actions of a group G on a set X are in bijection with homomorphisms from G to S_X.
• Check that conjugation defines a group homomorphism from G to Aut G.
• (bonus) Show that the automorphism group of Z x Z is isomorphic to GL_2(Z), the group of invertible two by two integer matrices. [NB: previous version was incorrect!]

Week 2:

• DF Chapter 2.2: 6,10,14
• 2.3: 1,11,24,26
• 2.4: 9,10,11
• 3.1: 1,12,14,16,32,36,41
• 3.2: 4, 11, 18
• 3.3: 2,9
• optional: 3.4: 7,9,10

Week 3:

• DF Chapter 4.1: 1,2,7
• 4.3: 19,20,27,29,30
• 4.4: 1,2,3,18
• Show that a solvable group is simple iff it is cyclic of prime order.

Week 4:

• DF Chapter 4.5: 3,4,13,16,17,19,21,23,30
• Suppose G is a group containing normal subgroups H,K such that HK=G and the intersection of H and K is {1}. Prove that G is isomorphic to HxK.

Week 5:

• DF 4.2: 11,12
• 4.5: 36,37,41
• 4.6: 1,3,4
• Let G be a group of order 2^m k, with k odd, and suppose G contains an element of order 2^m [correction - previously said 2^k..]. Show that the set of elements of G of odd order is a normal subgroup. Deduce (using the Feit-Thompson theorem) that a finite non-abelian simple group has order divisible by 4.
• Show that if |G|<100, |G| not 60, then G is solvable.

Week 6:

• DF 5.4: 7,9,13,19
• 5.5: 10, 11
• 6.1:8,21,25

Week 7

• 6.1: 7
• 7.1:7,9,10,13

Week 8

• 7.1:25, 26
• 7.2:3
• 7.3:12,17,22
• 7.4:2 (see p.245 for augmentation ideals), 13,18,37,38
• 8.2: 3,4

Week 9

• 9.2: 1,2,3,4
• 9.4: 2,10,12
• 15.1: 12,13
• 10.1:1,2,4,7,8,13,16

Week 10

• 10.2: 6,7,8,9
• 10.3:2,4,5,15,
• Let D be a division ring. a. Prove that D^{op} is a division ring. b. For any ring R prove that M_n(R)^{op}=M_n(R^{op}). c. Regarding R as a left R-module, show that there is a ring isomorphism End_R(R) = R^{op}.
• Suppose a left R-module M is isomorphic to the direct sum of n copies of a simple module L. Show that End_R(M) is isomorphic to Mat_n(End_R(L)).
• Prove that every submodule and every quotient module of a semisimple module is semisimple and the direct sum of semisimples is semisimple.
• The Jacobson Radical J of a ring R is the intersection of all maximal ideals in R. Compute the Jacobson radical of Z/30. Show in general that x is in J if and only if 1-xy is a unit in R for all y in R.

Week 11

• 18.1:3,8
• 10.5: 3,4,14,16
• Show that Q is an injective Z-module, but not a projective Z-module.
• Suppose R is an integral domain. Say an R-module is divisible if for each d in M and nonzero r in R, you can divide d by r (i.e. there's a d' in M such d=rd'). Show that any injective R-module is divisible.
• Let R be a PID, F its field of fractions, and D a finitely generated R-submodule of F. Show that D is cyclic.

Week 12

• (Due Thursday Dec.2)
• 12.1:17,18,19
• 12.2:1,3,4,10,15
• 12.3:21,22,31,32,38