Faithful Group Actions of a Group on a Set

Definition: Let $(G, \cdot)$ be a group and let $A$ be a set. Let $(g, a) \to ga$ be a (left) group action of the group $G$ on the set $A$. This group action is said to be Faithful if the corresponding associated permutation representation homomorphism $\varphi : G \to S_A$ 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 $\sigma_g(a) = ga$) is injective.

In other words, a left (or right) group action is faithful if $\varphi : G \to S_A$ is a group monomorphism.

We will now look at some examples of group actions that are faithful / not faithful.

Example 1

On The Trivial Group Action of a Group on a Set page we defined the trivial group action of a group $(G, \cdot)$ to a set $A$ for all $g \in G$ and for all $a \in A$ by $(g, a) \to a$. We noted that the corresponding associated permutation representation homomorphism $\varphi : G \to S_A$ is actually the homomorphism of $G$ onto the trivial subgroup $\{ \mathrm{id}_A \}$ of $S_A$.

If $(G, \cdot)$ is the trivial group then the the trivial group action of $G$ on the set $A$ is faithful.

If $(G, \cdot)$ is not the trivial group then the trivial group action of $G$ on the set $A$ is NOT faithful.

Example 2

On the The Left and Right Regular Group Actions of a Group on Itself page we defined the left regular group action of a group $(G, \cdot)$ to the set $A = G$ for all $g \in G$ and for all $a \in A$ by $(g, a) = g \cdot a$.

For each $g \in G$ let $\sigma_g : A \to A$ be defined for all $a \in A$ by $\sigma_g(a) = g \cdot a$. Let $\varphi : G \to S_A = S_G$ be the corresponding associated permutation representation homomorphism. We claim that the left regular representation of $G$ on the set $G$ is faithful, i.e., $\varphi$ is a group monomorphism.

Let $g_1, g_2 \in G$ and suppose that $\varphi(g_1) = \varphi(g_2)$. Then $\sigma_{g_1} = \sigma_{g_2}$. That is, for all $a \in A = G$ we have that:

(1)

\begin{align} \quad \sigma_{g_1}(a) &= \sigma_{g_2}(a) \\ \quad g_1 \cdot a &= g_2 \cdot a \end{align}

By the cancellation law for groups, we have that $g_1 = g_2$. So indeed, $\varphi$ is injective, so the left regular group action is faithful. A similar argument shows that the right regular group action is also faithful.

Example 3

On the A Group Action of the Symmetric Group SX on the Set X page we defined a group action of the symmetric group $S_X$ on the set $X$ for all $\sigma \in S_X$ and for all $x \in X$ by $(\sigma, x) \to \sigma(x)$.

We proved that if $\varphi : S_X \to S_X$ is the corresponding associated permutation representation homomorphism then $\varphi(\sigma) = \sigma$ for all $\sigma \in S_X$, i.e., $\varphi$ is the identity homomorphism. The identity homomorphism is certainly injective, and so this group action is faithful.