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).

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