# A note on faithful group actions

The following self-contained post clarifies the definition of faithful group action (definition 1.2, p. 3, in the book “Groups, graphs and trees: an introduction to the geometry of infinite groups” by John H. Meier).

Let $G$ be a group, $X$ a set and $\mbox{Sym}(X)$ the symmetric group of $X$. As it is known and as it has been elaborated in a previous post, a function $\phi : G \times X \rightarrow X$ is a group action if and only if the function $h:G\rightarrow \mbox{Sym}(X),~h(g)=f_g$, with $f_g(x)=\phi(g, x)=:gx$, is a group homomorphism. The purpose of the present post is to clarify the concept of faithful group action.

It is easy to show that $(\forall g\in G, g\ne e )(\exists x\in X)gx\ne x$ if and only if $(\forall g,s\in G, g\ne s )(\exists x\in X)gx\ne sx$. Either of these two equivalent statements defines a faithful group action $\phi$.

The faithful property of a group action $\phi$ is equivalent to properties of the associated group homomorphism $h$. More concretely, it is easy to show that a group action $\phi$ is faithful if and only if the associated group homomorphism $h$ is injective if and only if $h$ has a trivial kernel. Shortly, $\mbox{Ker}(h)=\left\{e\right\}\Leftrightarrow \phi~\mbox{faithful}\Leftrightarrow h~\mbox{injective}$, where $\mbox{Ker}(h)$ is the kernel of $h$ and $e$ the neutral element of $G$.

It can also be shown that if $\phi$ is injective, then $\phi$ is faithful. However, the converse does not hold. So, the well-known equivalence between injective and faithful actions states that a group action $\phi$ is faithful if and only if its associated group homomorphism $h$ is injective (whereas $\phi~\mbox{faithful}\Leftrightarrow\phi~\mbox{injective}$ does not hold).