Final exam Saturday 5/22/04, 8-11am.

[DF] means our book, Dummit and Foote.

Tues Jan 20	Proofs. Splitting into cases, proof by contradiction, proof by induction. Sets vs. lists. One common thing to prove: two sets are equal, which often must be done by two inequalities.
Thurs Jan 22	Definition of function. 1:1, onto, monic, epic. Theorem: monic = 1:1. Images, and how they constrain maps into a set. Partitions, and how they constrain maps out of a set. Definition of graphs and graph isomorphisms.
Tues Jan 27	Finishing up partitions and graph isomorphisms. Equivalence relations, as another description of partitions (see p3 of [DF], though there isn't much there).
Thurs Jan 29	Examples of graphs and their automorphism groups (including the n-gon and the cube). Definition of group, the notation x^n (that's my lame HTML way of indicating a superscript), and of the order of an element. Group tables. See p13-25 of [DF].
Tues Feb 3	Products of groups. "All" the groups of order at most 11 (statement only, since we haven't discussed group isomorphisms). Subgroups. Theorem: a nonempty subset of a finite group that is closed under multiplication is automatically a subgroup.
Thurs Feb 5	Permutation groups. The cycle structure of a permutation. Odd and even permutations. The Sam Loyd 14-15 puzzle is unsolvable. Orbits, as equivalence classes.
Tues Feb 10	Group homomorphisms, subgroups, group actions, stabilizers. Theorem: |G| = |orbit| |stabilizer|.
Thurs Feb 12	Examples of that last theorem. Recap for the midterm.
Tues Feb 17	Midterm #1
Thurs Feb 19	Lagrange's Theorem
Tues Feb 24	Conjugacy classes, the center, and groups of order p^2
Thurs Feb 26	Midterm recap
Tues Mar 2	Beginning normal subgroups -- why they're kernels of representations.
Thurs Mar 4	First isomorphism theorem. Simple groups.
Tues Mar 9	Euclidean algorithm. Subgroups of cyclic groups are cyclic. Fermat's Little Theorem (via a group action).
Thurs Mar 11	Euler's extension of FLT. Outer automorphism groups (just for fun).
Tues Mar 13	The Sylow subgroups of S_3 through S_6. If A normalizes B, then AB is a subgroup. If p^k || n, then {n choose p^k} is not a multiple of p. If G acts on X, and |X| is not a multiple of p, then some |G_x| is a multiple of p^k, where p^k || |G|. Sylow subgroups exist.
Thurs Mar 18	A better proof of p not dividing {n choose p^k}. The Sylow subgroups are all conjugate, and their number divides |G| / p^k. Moreover, it is congruent to 1 mod p. If there is a unique subgroup of a certain size, it is normal (important application: Sylow subgroups).
Tues Mar 30	Recognizing direct products. Definition of semidirect products, and how to recognize them. Classification of groups of order 6 & 10 using Sylow's theorems and semidirect products.

Coming next: more applications of Sylow. Classification of group actions. midterm #2.