Now that we know about group actions, I'm going to give a few examples of some of the more important group actions. I'm not going to prove all of them, but its a good exercise to prove them yourself if you're feeling ambitious.
Example:Trivial Action
Consider any group, $G$, and any set, $S$. Define the binary operation $G \times S \to S$ by $g \cdot s = s$ for $g \in G$ and $s \in S$. Then this operation defines a group action of $G$ on $S$ called the trivial action.
The trivial action is the most boring action that there is, but it is a group action. We don't really get anything exciting from the trivial action, except that it serves to show that any group acts on any set, which further emphasizes the fact that an action of a group on a set is defined by the particular operation.
In the last post, I mentioned that there were two important actions of groups on themselves, which I will give below. These actions are confusing to define, though, because the "group" and the "set" from the definition of group actions are the same thing.
Example:Regular Action of G
Let $G$ be any group and let $S = G$. Define the binary operation $G \times S \to S$ by $g \cdot s = g s$ for $g \in G$ and $s \in S$ where $g s$ is computed according to the operation of $G$ (which is legal since $S = G$). This operation defines a group action of $G$ on itself called the left regular action.
This is kind of the second most boring action we can define after the trivial action. It is a group action because it follows the rules of group actions, but its a really obvious group action. It may not seem like this is a useful thing to do because the regular action does not work any differently than the standard group operation. However, soon we're going to derive some useful theorems about group actions and the fact that the group operation defines a group action means that the theorems about group actions also apply to the group operation. The next action, though, is more interesting.
Example:Conjugation Action
Let $G$ be any group and let $S = G$. Define the binary operation $G \times S \to S$ by $g \cdot s = s g s^{-1}$ for $g \in G$ and $s \in S$ where $s g s^{-1}$ is computed according to the operation of $G$ (which is legal since $S = G$). This operation defines a group action of $G$ on itself and is called conjugation.
Proof:
Choose $g,h \in G$ and $x \in S$. Then we have
\[ (g h) \cdot x = (g h) x (g h)^{-1} = g h x h^{-1} g^{-1} = g (h \cdot x) g^{-1} = g \cdot (h \cdot x) \]
so that $(g h) \cdot x = g \cdot (h \cdot x)$.
Choose $x \in S$. Since $x \in S$ then we clearly have that $e \cdot x = e x = x$ by the definition of $e$.
Both of the conditions of a group action are satisfied, so the operation is a group action.
This conjugation action is very important. I don't want to underplay this. The conjugation action is extremely important. The conjugation action is important in the same way that LeBron James is important to the Cavaliers chances at an NBA title. The conjugation action is important in the same way that the Beatles and Elvis helped the success of rock music. I can't exactly explain why this is yet because we don't have the theorems, but as a sneak peak, conjugation partitions groups similar to the partition that we get from cosets except in a more interesting way. In the next few posts we'll learn some things that will help us eventually work some serious magic on the conjugation action.
I've got one last example that I'm going to throw in for fun, although it doesn't get used all that often.
Example:
Let $G$ be a group and suppose $N \triangleleft G$. For the purpose of this example, $G/N$ is going to be the set. Define the binary operation $G \times G/N \to G/N$ by $g \cdot h N = (g h) N$ for $g \in G$ and $h N \in G/N$. This operation defines a group action of $G$ on $G/N$.
Hopefully the examples cleared up group actions a little. Next we're going to learn about some of the various properties of group actions
References
Previous Related Post: Group Actions
Text Reference: Gallain Chapter 7 and Chapter 29
Wikipedia: Group Action
Wolfram Mathworld: Trivial Group Action
Wolfram Mathworld: Faithful Group Action
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