Trivial Group Action

trivial action

Suppose Group $G$ acts on $X$ . We say an element $g \in G$ acts trivially if $g x = x$ for all $x \in X$ .
The kernel of the Group Action is the set:

Ker (σ) = {g \in G | g x = x \forall x \in X} = {g \in G | σ_{g} = 1 \in S_{X}}

More generally, for any group morphism (map) $φ : G \to H$ the kernel is:

Ker (φ) := {g \in G | φ (g) = 1_{H}}

Example

The only element $g \in G$ that acts trivially is $1_{G}$ .
Proof

Take any $g \neq 1_{G}$ . If $x \in X$ is fixed by $g$ then:

g x = x ⟺ g = 1_{G}

A group $G$ can also act on itself by conjugation. Here $x \in G$ (confusing I know):

g \cdot x := g x g^{- 1}

Let's check that this is a valid Group Action.
a. $1_{G} \cdot x = 1_{G} x 1_{G}^{- 1} = 1_{G} x 1_{G} = x$
b. $g_{1} \cdot (g_{2} \cdot x) = g_{1} \cdot (g_{2} x g_{2}^{- 1}) = g_{1} (g_{2} x g_{2}^{- 1}) g_{1}^{- 1} = (g_{1} g_{2}) x (g_{2}^{- 1} g_{1}^{- 1}) = (g_{1} g_{2}) x (g_{1} g_{2})^{- 1} = g_{1} g_{2} \cdot x$ .

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