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Let $G$ a group acting on $X$. This is the set of all fixed points of $X$: $$ X^G = \{x\in X : g(x) = x, \forall g\in G\} $$

Let $T\subseteq X$ and $x\in X$. We know that, this one is the stabilizer, the fixed set, and the orbit: $$ G_x = Stab_G(x) = \{g\in G : g(x) = x\} \\ G_T = Fix_G(T) = \{g\in G : g(x) = x, \forall x\in T\} \\ G.x = G(x) = Orb_G(x) = \{g.x = g(x) : g\in G\} $$

Apparently wikipedia prefers the notation of $X^G$ and $G_X$ and $G_T$ and $G_x$ and $G.x$ and etc. Call it, notation $01$. But some other sources prefers $Stab_G(x)$ or $Fix_G(T)$ or $Orb_G(x)$, call it, notation $02$.

Question: Is there a notation $02$ for $X^G$? Say.. something like $FixedPoint_X(G)$? If there is, I couldn't find it anywhere...

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    I usually see $X^G$ instead of $X_G$. – 2017-02-28
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    @E.Joseph Oh! Apparently you are right. Wiki uses $X^g$. Well.. But I also saw $X_G$ somewhere.... – 2017-02-28
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    In a context where $X^G$ can be mistaken with the set of functions from $G$ to $X$, maybe it is preferable to use $X_G$ instead. But I think otherwise that $X^G$ is a pretty standard notation. – 2017-02-28
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    If you truly can't find anything suitable, and you find you really do need alternative notation, you can always make something up. – 2017-03-01
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    $X_G$ can be used to mean the set of $G$-orbits. – 2017-03-01

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