= Holomorph
{title2=$\operatorname{Hol}(G)=G\rtimes\operatorname{Aut}(G)$}
= Holomorph of a group
{synonym}
The holomorph acts on the underlying set of a <group> by $x\mapsto g\alpha(x)$. Its multiplication is $(g,\alpha)(h,\beta)=(g\alpha(h),\alpha\beta)$. This is exactly the <normalizer> of the left regular <permutation> representation: a normalizing <permutation> fixing the identity is an <group automorphism>, and every normalizing <permutation> is a left translation followed by one. The embedded copy of $G$ is normal and the identity stabilizer is its <group automorphism> <group>.
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