MAT 541

In this first lecture we discussed in broad terms the concept of symmetry: rearrangemnet of objects on a table, or the rigid motion of objects in 3 dimensional space, or the plane. We discovered the Euclidean group (in particular, the notion of a group ) and a non-obvious "multiplication" of such motions. This will later be christened as the semi-direct product.

The most delicate point in this lecture was the concept of an "equivalence" class and the "reduction" of a set X modulo an equivalence relation. The basics of sets, and mappings were also reviewed.

We will look into more examples of equivalence relations. We will discuss the euclidean algorithim and the "principle of inclusion exclusion", a useful counting technique.

We will finish the principle of inclusion/exclusion. Discuss the euclidean algorithm, and begin the basics of group theory.

We defined an abstract group. We discussed several examples of groups, in particular we discussed the quaternion group.

We discussed various ways of writing down permutations: two line notation, one line notation, and cycle notation. We started discussing integers modulo n.

The most important concept of todays lecture is that of a homomorphism (and isomorphism) between two groups. We showed that integers mod 2 and {1,-1} are isomorphic as groups.

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Basic definitions. The symmetric group on 3 letters, D_6, and GL(2,F_2) are all isomorphic. Key was the notion of a group acting on a set.

Basic definitions. The symmetric group on 3 letters, D_6, and GL(2,F_2) are all isomorphic. Key was the notion of a group acting on a set.

Notion of a group acting on a set. We began to discuss the significant example of the automorphisms of the unit disk in the complex plane. The richness of the topic can be attributed to complex analysis and hyperbolic geometry (which are outside of the sope of this course).

Definitions and basic properties of the action of SU(1,1) on D (the unit disk).

We discussed cosets with respect to a subgroup H of a group G. When H is normal the set of cosets is a group in a natural way.

More on quotients and homorphisms.

We introduce a beautiful transformation from the disk to the upper half plane discovered by Arthur Cayley. This will show in a natural way that PSU(1,1) and PSL(2,R) are isomorphic.

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