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What is a stabilizer of a set?

I know what a stabilizer of $x\in X$ with respect to a group $G$ that acts on $X$ is, specifically: $$\{g\in G:g\cdot x=x\}.$$

But when defining the stabilizer of a set $Y\subset X$ this could go two ways: $$\{g\in G:g\cdot x=x,\quad \forall x\in Y\},$$ $$\{g\in G:g\cdot Y\subset Y\}.$$

Which is it? I couldn't find online the definition.

The later allows for example that if we had $Y=\{a,b\}$ that $g\cdot a =b, g\cdot b =a$ is a viable element of the latter 'stabilizer'.

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    They are called respectively the *pointwise* and *setwise* stabilizers of the set. But for the setwise stabilizer you want $\{ g \in G : g \cdot Y = Y \}$. – 2017-01-23

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If a group $G$ acts on a set $\Omega$, we may extend this to an action of $G$ on the set of all subsets of $\Omega$ (its power set).

This is done by declaring for $S \subseteq \Omega$ that $g \cdot S = \{g \cdot s : s \in S\} \subseteq \Omega$.

In this case, the stabilizer of a subset $S$ is any group element that fixes $S$ as a subset, not necessarily fixing each $s \in S$. In the latter case, when $g \cdot s = s$ for all $s \in S$, we say that $g$ fixes $S$ pointwise (I guess we could say "stabilizes pointwise" but it's much less common in my experience). In general, stabilizers of subsets may permute elements within the subset.

If I hear something about the stabilizer of a subset, I automatically think "not necessarily pointwise" unless it is explicitly mentioned; I don't think I'm in the minority. But if I'm doing the writing, I'll always explicitly mention whether things are fixed pointwise or not, because the extra clarification never hurts.

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    No I don't think you are in the minority but unfortunately (?) one of the most widely read and most influential books on permutation groups, the one by Wielandt, uses the notation $G_S$ for the pointwise stabilizer. – 2017-01-23
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    @DerekHolt Thank you for pointing that out. I was afraid the story wouldn't be that simple. It seemed safe to assume that "*the* stabilizer" of a set would be the setwise one, since it seems to be what the definition of *stabilizer* would imply for the usual induced action. But, I guess the lesson here is that it's just not safe to refer to *the* stabilizer of a subset without any further qualifications. – 2017-01-23