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 be a group, a set and the symmetric group of . As it is known and as it has been elaborated in a previous post, a function is a group action if and only if the function , with , 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 if and only if . Either of these two equivalent statements defines a faithful group action .
The faithful property of a group action is equivalent to properties of the associated group homomorphism . More concretely, it is easy to show that a group action is faithful if and only if the associated group homomorphism is injective if and only if has a trivial kernel. Shortly, , where is the kernel of and the neutral element of .
It can also be shown that if is injective, then is faithful. However, the converse does not hold. So, the well-known equivalence between injective and faithful actions states that a group action is faithful if and only if its associated group homomorphism is injective (whereas does not hold).