TITLE: What is difference between free and faithful group action?
QUESTION [10 upvotes]: I searched Wikipedia for definitions of free and faithful actions. As I understand them, the two concepts are the same thing!
If they are one concept, what is the point of introducing both or even of naming them in distinct ways?

REPLY [9 votes]: As Keith Conrad explains on his notes:

"It is worth comparing faithful and free actions. An action is faithful when ${g_1 \neq g_2  \Rightarrow  g_1 x \neq g_2 x}$ for some ${x \in X}$ (different elements of $G$ act differently at some point) while an action is free when ${g_1 \neq g_2 \rightarrow g_1 x \neq g_2 x}$ for all ${x \in X}$ (different elements of $G$ act differently at every point). Since ${g_1 x = g_2 x}$ if and only if $g_2^{-1} g_1 x = x$ we can describe faithful and free actions in terms of fixed points: an action is faithful when each $g \neq e$ has Fix${_g}(X) \neq X$ while an action is free when each $g \neq e$ has Fix${_g}(X) = \emptyset$."