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 $G$-action, G-invarient, G-equivarient]

• Let $G$ be a group, $X$ be a set. A left $G$-action on $X$ is a map $*:G \times X \rightarrow X$ such that:

  1. For any $x \in X$, $*(e_G, x) = x$;
  2. For any $x \in X$, $a, b \in G$, $*(a, *(b, x)) = *(ab, x)$. Then the pair $(X, *)$ is called a left-$G$ set.

  When there's no ambiguity, we write $*(a, b)$ as $a*b$ or $ab$.

• A subset $Y \subseteq X$ is called $G$-invarient if $\forall y \in Y$, $\forall g \in G$, we have $y*g\in Y$;

• Let $(X, *)$, $(Y, *')$ be two left $G$-sets. A map $f: X \rightarrow Z$ is called $G$-equivarient if $\forall x \in X$, $\forall g \in G$, $f(g*x) = g*f(x)$.

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. :(