In order to examine the characteristics of center group spaces, center orbits, center stabilizers, and center kernels, the major purpose of this work is to introduce the definition of center group actions. We review the definition and proposition of the center limit set, as well as the center thin sets (briefly C{\mathscr{C}}-thin) and the center Cartan Cp(G){Cp}\left(G)-spaces (briefly C{\mathscr{C}}-Cartan Cp(G){Cp}\left(G)-spaces).
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