Monday, April 7 (Day 19): Proved Euler's theorem and Fermat's little theorem as corollaries of Lagrange's theorem. Introduce the notion of an isomorphism and isomorphic groups and discussed two examples.
Wednesday, April 9 (Day 20): Discussed several basic properties of isomorphisms and isomorphic groups. Classified cyclic groups, that is, we proved that two cyclic groups are isomorphic if and only if they have the same cardinality.
Wednesday, April 2 (Day 18): Proved Lagrange's theorem, discussed some corollaries. Showed that the converse of Lagrange's theorem is false by showing that the alternating group on four letters did not have a subgroup of order six. (Class had a sub.)
I will be away Wednesday, April 2; class will be taught by Junjie Chen, a PhD student at the CUNY Graduate Center.
Monday, March 24 (Day 16): Finished our discussion of permutation groups, including a discussion of the order of the alternating group and how to compute the order of a permutation. Formally introduced the dihedral groups as automorphism groups of graphs. Computed the order of the dihedral groups and listed and described their elements.
Wednesday, March 26 (Day 17): Discussed how to realize the dihedral groups as subgroups of symmetric groups. Introduced the notions of left and right cosets, saw some examples, and proved a lemma giving equivalent conditions for two cosets to be equal. Proved that the left (resp., right) cosets of a subgroup partition the group. Discussed the relationship between partitions and equivalence relations, and how the partitioning theorem just mentioned is equivalent to an old homework problem. (Class was interrupted by a fire alarm.)
Week 8
Reading: No reading this week.
Monday, March 17 (Day 14): Class cancelled (I was sick).
Office hours on Monday, March 10 are 1:30–2:30pm (instead of starting at 1pm).
Monday, March 10 (Day 12): Recalled the definition of a k-cycle. Proved that every permutation is either a cycle or can represented as a product of disjoint cycles.
Wednesday, March 12 (Day 13): Introduced the notion of even and odd permutations, and proved that these notions are well defined. Introduced the alternating subgroup, that is, the subgroup consisting of even permutations.
We have class on Thursday (CUNY is on a Wednesday schedule).
Monday, March 3 (Day 9): Proved a second subgroup test. Introduced cyclic groups, cyclic subgroups, and the cyclic subgroup generated by an element. Defined the order of an element. Went over examples of all the above.
Wednesday, March 5 (Day 10): Established several properties of cyclic groups. Proved that cyclic groups are abelian and that every subgroup of a cyclic group is cyclic. We are working towards computing the order of an element in a cyclic group.
Thursday, March 6 (Day 11): Introduced permutation groups, and began discussing how to think about and how to represent permutations of the set {1,2,...,n}.
Homework 2 is complete as of Day 7 (Mon., Feb. 24).
We have class on Thursday, March 6.
Monday, Feb. 24 (Day 7): Established basic properties of groups, e.g., uniqueness of identity, uniqueness of inverses, formula for the inverse of a product, and right and left cancellation laws.
Wednesday, Feb. 26 (Day 8): Introduced the notion of a subgroup and discussed several examples. Discussed the notion of a group invariant and used the number of index 2 subgroups to distinguish two groups. Went over the subgroup test.
Tuesday, Feb. 18 (Day 5): Discussed multiplicative inverses in modular arithmetic. Introduced Cayley tables and rigid motions. Began exploring the rigid motions of the equilateral triangle.
Wednesday, Feb. 19 (Day 6): Made the Cayley table for the rigid motions of the equilateral triangle. Introduced the definition of a group and went over several examples. (Note: the problems on HW2 have gotten ahead of the class; in particular, we have not covered the necessary material for #17–21.)
No class Wednesday, Feb. 12 and Monday, Feb. 17 (College closed).
We have class on Tuesday, Feb. 18. We have Test #1 on this day.
Monday, Feb. 10 (Day 4): Showed how to write gcd as linear combination using the Euclidean algorithm.
Introduced equivalence modulo n and modular arithmetic.
Homework 1 is now complete (as of Feb. 5); it covers the material from Days 1–3 of class.
Monday, Feb. 3 (Day 2): Proved the division algorithm, introduced greatest common divisor (gcd), and established the Euclidean algorithm.
Wednesday, Feb. 5 (Day 3): Proved that the gcd is a linear combination, introduced primes, proved Euclid's lemma, proved that every natural number greater than one is a product of prime numbers, proved there are infinitely many primes, and stated the fundamental theorem of arithmetic.