# What does it mean for an automorphism to centralize a subgroup?

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?

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