Alg7.9

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Define actions, centralizers, normalizers, stabilizers, and centers. g\isin G\,, and x\isin 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\isin G\bigg|\, gxg^{-1}\isin X,\,\forall x\isin X\right\}\,

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

  • Z_G\triangleleft G\,

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

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

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

Abstract Algebra

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