The Trivial Group Action of a Group on a Set

# The Trivial Group Action of a Group on a Set

Recall from the Group Actions of a Group on a Set page that if $(G, \cdot)$ is a group and $A$ is a (nonempty) set then a left group action of $G$ on the set $A$ is a map $G \times A \to A$ denoted for all $g \in G$ and for all $a \in A$ by $(g, a) \to ga$ that satisfies the following properties:

• 1) $g_1(g_2a) = (g_1 \cdot g_2)a$ for all $g_1, g_2 \in G$ and for all $a \in A$.
• 2) $1a = a$ for all $a \in A$ (where $1 \in G$ denotes the identity).

We will now look at the simplest group action of a group $G$ on a set $A$ - the trivial group action.

 Definition: Let $(G, \cdot)$ be a group and let $A$ be a (nonempty) set. The Trivial (Left & Right) Group Action of $G$ on the set $A$ is the left group action defined for all $g \in G$ and for all $a \in A$ by $(g, a) \to a$.

We should indeed verify that the trivial group action is indeed a group action of $G$ on the set $A$.

For all $g_1, g_2 \in G$ and for all $a \in A$ we have that:

(1)
\begin{align} \quad g_1(g_2a) = g_1a = a = (g_1 \cdot g_2)a \end{align}

For all $a \in A$ we have that:

(2)
\begin{align} \quad 1a = a \end{align}

So indeed, $(g, a) \to a$ is a left group action of $G$ on the set $A$. Similarly, it is a right group action of $G$ on the set $A$.

Let $\varphi : G \to S_A$ be the associated permutation representation homomorphism defined for all $g \in G$ by $\varphi(g) = \sigma_g$ where $\sigma_g : A \to A$ is defined for all $a \in A$ by $\sigma_g(a) = ga = a$. Note that for each $g \in G$ we have that $\sigma_g = \mathrm{id}_A$. Thus for all $g \in G$ we have that:

(3)
\begin{align} \quad \varphi : g \to \mathrm{id}_A \end{align}

So $\varphi : G \to S_A$ is the trivial homomorphism of the group $G$ to the trivial subgroup $\{ \mathrm{id}_A \}$ of $S_A$.