Suppose $H$ is a subgroup of a group $G$. What does it mean for an automorphism $\sigma\colon G\to G$ to centralize $H$?

The context I have is that if $\sigma$ centralizes $H$, then the map $x\mapsto \sigma(x)x^{-1}$ sends $H$ to the identity, which would mean $\sigma(h)=h$ for $h\in H$, so does it mean $\sigma$ fixes $H$ pointwise, or something more general?

  • 2
    $\begingroup$ Yes, it simply means that $\sigma$ fixes $H$ pointwise. $\endgroup$ – Dan Shved Jan 10 '16 at 7:30
  • $\begingroup$ @DanShved Thanks. $\endgroup$ – gnuck Jan 10 '16 at 7:31

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