I was in class today, and the teacher mentioned that the semidirect product given by $N \rtimes \textrm{Aut}(N)$ has a special name, but I couldn't understand what he said well enough to write it down. Does anyone know what it's called, what it's useful for, and where I could learn more about it?
2026-03-25 18:51:10.1774464670
What's the special name for $N \rtimes \textrm{Aut}(N)$?
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This type of semidirect product is the so-called holomorph of the group $N$ and (at least the name) was introduced by Burnside. Apparently (after a quick google search), the following problem/question is the origin of the holomorph: "Is it possible to include any given group as a normal subgroup in some other group so that all the automorphisms of $N$ are restrictions of inner automorphisms of this large group?"
See here for more information and more uses.