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I think that in "Group Action" section ${\displaystyle \rho :G\to \mathrm {Aut} (X)}$ should be replaced by ${\displaystyle \rho :G\to \mathrm {Sym} (X)}$, because there is not (necessarily) a group structure/operation on X. Also in the "Group Action" article the talk is about induced permutations, thus symmetric group. But I'm not an expert to correct it.