Group Action

I took Abstract Algebra in UW-Madison(MATH541) with professor Chenxi Wu. This article is inspired by prof. Wu's lecture note, and many formats, content, and structure of this blog are influenced by and similar to it.

Definition 1. [Left -action, G-invarient, G-equivarient]

• Let be a group, be a set. A left -action on is a map such that:
  1. For any , ;
  2. For any , , . Then the pair is called a left- set.
  When there's no ambiguity, we write as or .
• A subset is called -invarient if , , we have ;
• Let , be two left -sets. A map is called -equivarient if , , .

Definition 2. [Left action, Conjugate action]

Since I just found that there's likely no group theory in the final exam, I stop updating this part until final. :(