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Define actions, centralizers, normalizers, stabilizers, and centers. g\in G\,, and x\in X\,is fixed.

If a set G\, acts on a set X\, then there exist homomorphisms G\rightarrow S_{X}\, where S_{X}\, is the group of permutations of the elements of G\,.

Normalizer of X\, in G\,: N_{G}(X)=\left\{g\in G{\bigg |}\,gxg^{{-1}}\in X,\,\forall x\in X\right\}\,

Centralizer of x\in X\, in G\,: C_{G}(x)=\left\{g\in G{\bigg |}\,gx=xg\right\}\,

  • C_{G}(s)<G\,

  • C_{G}(s)<N_{G}(s)<G\,

Center of G\,: Z(G)=C_{G}(G)=\left\{g\in G{\bigg |}gx=xg\,\forall x\in G\right\}\,

  • Z_{G}\triangleleft G\,

Stabilizer of x\in X in G\,: G_{x}=\left\{g\in G{\bigg |}\,g\cdot x=x\right\}\,

  • If S\subset G\, then for action by conjugation G_{S}=\left\{g\in G{\bigg |}\,gSg^{{-1}}=S\right\}=N_{G}(S)\,.

Orbit of x\in X in G\,: O_{x}=\left\{g\cdot x{\bigg |}\,g\in G\right\}\,

Abstract Algebra

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