# Fully normalized subgroup

In symbols, a subgroup ${\displaystyle H}$ is fully normalized in ${\displaystyle G}$ if, given an automorphism ${\displaystyle \sigma }$ of ${\displaystyle H}$, there is a ${\displaystyle g\in G}$ such that the map ${\displaystyle x\mapsto gxg^{-1}}$, when restricted to ${\displaystyle H}$ is equal to ${\displaystyle \sigma }$.