Notations for Orbits, Stabilizers, Centralizers, Conjugation Classes, and the Center of a group

Question

I'm trying to understand these different aspects of group actions and the thing that's throwing me off the most is the notation for these sets. Additionally, when browsing other stack exchange questions and wikipedia articles, the notions aren't grouped together so it I can't compare one notation to another. Could someone please provide the common notations for the above sets to help clarify what notation goes with what? Thank you.

Answer 1

The orbit of a group action of $G$ on an element $x$ of a set $X$ is often denoted by $$G_X(x).$$ (See, for example, here.)

The stabiliser of an element $x$ of a set $X$ by a group action of $G$ is often denoted by $$\operatorname{Stab}_G(x).$$ (See, for example, here.)

The centraliser of a subset $S$ of a group $G$ is denoted by $$C_G(S).$$ (See, for example, here.)

The conjugacy class of an element $g$ of a group $G$ is often denoted $$\operatorname{Cl}_G(g).$$ (See, for example, here.)

The centre of a group $G$ is denoted by $$Z(G).$$ This is universal notation and can be found anywhere.