TITLE: Is the definition of stabilizer given at Planet Math really the currently accepted definition among group theorists?
QUESTION [8 upvotes]: According to Planet Math, given

a group $G$
a set $X$
a subset $S \subseteq X$
and a group action $G \times X \rightarrow X,$

then the stabilizer of $S$ is define to be: $\{g \in G \mid gS \subseteq S\}.$
I have an issue with this definition; I think the condition $gS \subseteq S$ should be replaced by $gS = S.$ (Essentially, wikipedia only defines stabilizers when $S$ is a a singleton set, thereby avoiding this issue. nLab also goes this way.)
To see the problem, let $X$ denote the real line, and write $G$ for the group of automorphisms of $X$ (i.e. order-preserving bijections from $X$ back to itself). Let $S = [-1,1]$. Then according to Planet Math, the function
$$g : x \in X \mapsto x/2 \in X$$
is in the stabilizer of $S$. But this seems odd, because the restriction of the above function to a mapping $S \rightarrow S$ does not yield a bijection. The problem, of course, is that $gS = [-1/2,1/2]$ is a proper subset of $S = [-1,1]$. Anyway, my question is:

Question. Is the definition of stabilizer given at Planet Math really the currently accepted definition among group theorists?

REPLY [4 votes]: It is a typo or sloppiness, since David Jao claims that the stabilizer is a subgroup, and it is not a subgroup with his "definition." The correct definition would require the equality, not the containment. I like the French terminology, where the "fixator" is used for the pointwise stabilizer of the subset.