Visual Group Theory, Lecture 5.1: Groups acting on sets

Visual Group Theory, Lecture 5.1: Groups acting on sets When we first learned about groups as collections of actions, there was a subtle but important difference between actions and configurations. This is the tip of the iceberg of a more general and powerful concept of a group action. Many deep results in group theory have clever proofs using a seemingly related group action. Formally, a group action arises when there is a homomorphism from G to the group Perm(S) of permutations of a set S. One can think of this as a "group switchboard": every element has a button that permutes things in S, with the requirement that "pressing the a-button followed by the b-button is the same as pressing the ab-button". We also see an alternative way to formalize this, and examine the subtle difference between left and right group actions, with plenty of visual examples to motivate the concepts. Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/math4120-online.html