📂Abstract Algebra

group Actions

Definition ¹

An action of a group $G$ with identity element $e$ on a set $X$ is a binary operation $\ast : G \times X \to X$ that satisfies the following two conditions:

• (i): For all $x \in X$, there is $ex = x$
• (ii): For all $x \in X$ and $g_{1} , g_{2} \in G$, there is $( g_{1} g_{2} ) (x) = g_{1} (g_{2} x)$

We refer to $X$ as a $G$-set under these circumstances.

Explanation

In simple terms, the action of a group is ‘adding $g \in G$ to $x \in X$’. To intuitively understand this, consider the following illustration:

$$ X : = \left\{ C, 1,2,3,4 , p_{1}, p_{2} , p_{3} , p_{4} , s_{1}, s_{2} , s_{3} , s_{4} , d_{1}, d_{2} , m_{1} , m_{2} \right\} $$ Regarding the set $X$, consider the tetrahedral group $D_{4}$. The set of lines and points that can be thought of in a square, $X$, can change its position through flipping and rotating by $D_{4}$, making it a $D_{4}$-set. Calling such manipulations that induce changes in $X$ an action is very reasonable and justifiable.

Note that $X$ doesn’t necessarily have to be a group nor related to $G$. For instance, considering $\mathbb{Z}$ and $$ X:= \left\{ \cdots , - {{3} \over {2}} , - {{1} \over {2}}, {{1} \over {2}} , {{3} \over {2}} , \cdots \right\} $$ $\left< X , + \right>$ doesn’t form a group and is not yet related to $G = \mathbb{Z}$. However, if $\ast : \mathbb{Z} \times X \to X$ is defined as $ z * x = z + x$ for $z \in \mathbb{Z}$ and $x \in X$, then

• (i): $0 + x = x$ holds,
• (ii): $(z_{1} + z_{2}) + x = z_{1} + (z_{2} + x)$ holds, hence the operation $\ast$ becomes an action on $X$, making $X$ a $\mathbb{Z}$-set.