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What is the motivation/intuition behind these concepts? What notion/property of a group do they capture? Or what is the scenario of application.

Thanks.

  • 3
    The centralizer of an element and the normalizer of a subgroup are both special cases of the stabilizer of a point for a group action. When $G$ acts on $G$ by conjugation, the stabilizer of a point in $G$ is its centralizer in $G$. When $G$ acts on its subgroups by conjugation, the stabilizer of a subgroup $H$ is its normalizer in $G$. Applications: the size of the conjugacy class of $g$ is the index of the centralizer of $g$, and the number of subgroups of $G$ that are conjugate to $H$ is the index of the normalizer of $H$ in $G$. Normalizers of Sylow subgroups appear in the 3rd Sylow thm.2012-12-23
  • 0
    http://math.stackexchange.com/a/215146/12952 Check out my answer here.2012-12-23

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