189-456A: Honors Algebra 3

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Lecture 1, on Wednesday August 30 was somewhat marred by a mix-up in classrooms which caused the instructor to arrive 20 minutes late. We discussed the axioms for a group and motivated the group concept with the principle that groups are an abstract way to characterize the symmetries of a (mathematical) object.
The relevant reading for this week and the coming ones are chapters 3 to 6 of Judson, and Chapter 2 of Garrett. The latter is more faithful to the way I will present the material in the class, while the former provides a more gentle exposition at a somewhat more leisurely pace.

Lecture 2, on Friday September 1 will describe a number of examples of groups, and then discuss subgroups and cosets, culminating in the proof of Lagrange's theorem, which asserts that the cardinality of every subgroup of a finite group divides the order of that group. As preparation for this, you may want to go through Chapter 2.2 of Garrett.

Lecture 3, on Wednesday September 6 discussed the notion of an action of a goup on a mathematical structure, with emphasis on the simplest case of sets. An important class of examples of a set with an action of a group G is the set G/H of cosets of a subgroup H in G, equipped with the action of G given by left multiplication. The fact that these cosets determine a partition of G into a disjoint union of #G/H subsets of cardinality #H leads to the important formula #G = #H . #(G/H) which implies, in particular, Lagrange's theorem that the cardinality of a subgroup always divides the cardinality of the ambient group.

Lecture 4, on Friday September 8. We start with the proof that every transitive G-set is isomorphic, as a G-set, to a G-set of the form G/H for a suitable subgroup H of G, which is well defined up to conjugation in G. We then gave a number of examples of sets on which G acts, illustrating how the classification of these leads to useful insights into the structure of groups. Next week, a particularly striking application of these ideas will be given as we exploit them to prove Sylow's theorems in group theory.